Theorem infmap2 7541

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. We start with infmap2lem2 7540 and also prove the other direction of the dominance relation. We obtain equinumerosity with Schroeder-Bernstein sbth 4446 and finally eliminate the degenerate case B = (/).

Hypotheses

Ref	Expression
infmap2.1	|- A e. V
infmap2.2	|- B e. V

Assertion

Ref	Expression
infmap2	|- ((om ~<_ A /\ B ~<_ A) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})

Distinct variable groups:   x,A   x,B

Proof of Theorem infmap2

Step	Hyp	Ref	Expression
1		opreq2 3964	. . 3 |- (B = (/) -> (A ^m B) = (A ^m (/)))
2		breq2 2619	. . . . 5 |- (B = (/) -> (x ~~ B <-> x ~~ (/)))
3	2	anbi2d 615	. . . 4 |- (B = (/) -> ((x (_ A /\ x ~~ B) <-> (x (_ A /\ x ~~ (/))))
4	3	abbidv 1575	. . 3 |- (B = (/) -> {x | (x (_ A /\ x ~~ B)} = {x | (x (_ A /\ x ~~ (/))})
5	1, 4	breq12d 2627	. 2 |- (B = (/) -> ((A ^m B) ~~ {x | (x (_ A /\ x ~~ B)} <-> (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))}))
6		infmap2.1	. . . . . . . 8 |- A e. V
7		infmap2.2	. . . . . . . 8 |- B e. V
8	6, 7	infxpabs 7530	. . . . . . 7 |- ((om ~<_ A /\ B =/= (/) /\ B ~<_ A) -> (A X. B) ~~ A)
9	8	3com23 838	. . . . . 6 |- ((om ~<_ A /\ B ~<_ A /\ B =/= (/)) -> (A X. B) ~~ A)
10	9	3expa 832	. . . . 5 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A X. B) ~~ A)
11	7, 6	xpcomen 4428	. . . . . 6 |- (B X. A) ~~ (A X. B)
12		entrt 4404	. . . . . 6 |- (((B X. A) ~~ (A X. B) /\ (A X. B) ~~ A) -> (B X. A) ~~ A)
13	11, 12	mpan 694	. . . . 5 |- ((A X. B) ~~ A -> (B X. A) ~~ A)
14	10, 13	syl 10	. . . 4 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (B X. A) ~~ A)
15	7, 6	xpex 3256	. . . . 5 |- (B X. A) e. V
16	15, 6	ssenen 4493	. . . 4 |- ((B X. A) ~~ A -> {x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)})
17		oprex 3978	. . . . . 6 |- (A ^m B) e. V
18		abid2 1578	. . . . . . 7 |- {x | x e. (A ^m B)} = (A ^m B)
19	6, 7	elmap 4327	. . . . . . . . 9 |- (x e. (A ^m B) <-> x:B-->A)
20		fssxp 3632	. . . . . . . . . 10 |- (x:B-->A -> x (_ (B X. A))
21		ffun 3625	. . . . . . . . . . . 12 |- (x:B-->A -> Fun x)
22		visset 1810	. . . . . . . . . . . . 13 |- x e. V
23	22	fundmen 4418	. . . . . . . . . . . 12 |- (Fun x -> dom x ~~ x)
24	22	ensym 4402	. . . . . . . . . . . 12 |- (dom x ~~ x -> x ~~ dom x)
25	21, 23, 24	3syl 20	. . . . . . . . . . 11 |- (x:B-->A -> x ~~ dom x)
26		fdm 3627	. . . . . . . . . . 11 |- (x:B-->A -> dom x = B)
27	25, 26	breqtrd 2635	. . . . . . . . . 10 |- (x:B-->A -> x ~~ B)
28	20, 27	jca 288	. . . . . . . . 9 |- (x:B-->A -> (x (_ (B X. A) /\ x ~~ B))
29	19, 28	sylbi 199	. . . . . . . 8 |- (x e. (A ^m B) -> (x (_ (B X. A) /\ x ~~ B))
30	29	ss2abi 2117	. . . . . . 7 |- {x | x e. (A ^m B)} (_ {x | (x (_ (B X. A) /\ x ~~ B)}
31	18, 30	eqsstr3 2089	. . . . . 6 |- (A ^m B) (_ {x | (x (_ (B X. A) /\ x ~~ B)}
32		ssdomg 4398	. . . . . 6 |- ((A ^m B) e. V -> ((A ^m B) (_ {x | (x (_ (B X. A) /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)}))
33	17, 31, 32	mp2 43	. . . . 5 |- (A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)}
34		domentr 4411	. . . . 5 |- (((A ^m B) ~<_ {x | (x (_ (B X. A) /\ x ~~ B)} /\ {x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)}) -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
35	33, 34	mpan 694	. . . 4 |- ({x | (x (_ (B X. A) /\ x ~~ B)} ~~ {x | (x (_ A /\ x ~~ B)} -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
36	14, 16, 35	3syl 20	. . 3 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)})
37		eqid 1474	. . . . 5 |- {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)} = {<.z, w>. | ((z (_ A /\ z ~~ B) /\ w:B-onto->z)}
38	6, 7, 37	infmap2lem2 7540	. . . 4 |- {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)
39		sbth 4446	. . . 4 |- (((A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)} /\ {x | (x (_ A /\ x ~~ B)} ~<_ (A ^m B)) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
40	38, 39	mpan2 695	. . 3 |- ((A ^m B) ~<_ {x | (x (_ A /\ x ~~ B)} -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
41	36, 40	syl 10	. 2 |- (((om ~<_ A /\ B ~<_ A) /\ B =/= (/)) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})
42		1onn 4246	. . . . . 6 |- 1o e. om
43	42	elisseti 1815	. . . . 5 |- 1o e. V
44	43	enref 4381	. . . 4 |- 1o ~~ 1o
45	6	map0e 4335	. . . 4 |- (A ^m (/)) = 1o
46		df-sn 2409	. . . . 5 |- {(/)} = {x | x = (/)}
47		df1o2 4133	. . . . 5 |- 1o = {(/)}
48		en0 4413	. . . . . . . 8 |- (x ~~ (/) <-> x = (/))
49	48	anbi2i 480	. . . . . . 7 |- ((x (_ A /\ x ~~ (/)) <-> (x (_ A /\ x = (/)))
50		0ss 2298	. . . . . . . . 9 |- (/) (_ A
51		sseq1 2079	. . . . . . . . 9 |- (x = (/) -> (x (_ A <-> (/) (_ A))
52	50, 51	mpbiri 194	. . . . . . . 8 |- (x = (/) -> x (_ A)
53	52	pm4.71ri 637	. . . . . . 7 |- (x = (/) <-> (x (_ A /\ x = (/)))
54	49, 53	bitr4 176	. . . . . 6 |- ((x (_ A /\ x ~~ (/)) <-> x = (/))
55	54	abbii 1573	. . . . 5 |- {x | (x (_ A /\ x ~~ (/))} = {x | x = (/)}
56	46, 47, 55	3eqtr4r 1504	. . . 4 |- {x | (x (_ A /\ x ~~ (/))} = 1o
57	44, 45, 56	3brtr4 2639	. . 3 |- (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))}
58	57	a1i 8	. 2 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m (/)) ~~ {x | (x (_ A /\ x ~~ (/))})
59	5, 41, 58	pm2.61ne 1631	1 |- ((om ~<_ A /\ B ~<_ A) -> (A ^m B) ~~ {x | (x (_ A /\ x ~~ B)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  {cab 1462   =/= wne 1583  Vcvv 1808   (_ wss 2044  (/)c0 2277  {csn 2406   class class class wbr 2615  {copab 2662  omcom 3127   X. cxp 3164  dom cdm 3166  Fun wfun 3172  -->wf 3174  -onto->wfo 3176  (class class class)co 3958  1oc1o 4121   ^m cm 4315   ~~ cen 4357   ~<_ cdom 4358

This theorem is referenced by:  alephexp2 7546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-iso 3195  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-2o 4127  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-map 4317  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-n 5883  df-2 5927  df-n0 6057  df-z 6093  df-seq1 6258  df-exp 6514

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