Theorem pwfi 4545

Description: The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105.

Assertion

Ref	Expression
pwfi	|- (E.n e. om A ~~ n <-> E.n e. om P~A ~~ n)

Distinct variable group:   A,n

Proof of Theorem pwfi

Step	Hyp	Ref	Expression
1		breq2 2613	. . . 4 |- (n = m -> (A ~~ n <-> A ~~ m))
2	1	cbvrexv 1792	. . 3 |- (E.n e. om A ~~ n <-> E.m e. om A ~~ m)
3		visset 1804	. . . . . . . 8 |- m e. V
4	3	pwen 4483	. . . . . . 7 |- (A ~~ m -> P~A ~~ P~m)
5	3	pwex 2735	. . . . . . . 8 |- P~m e. V
6		enen1 4457	. . . . . . . 8 |- ((P~m e. V /\ P~A ~~ P~m) -> (P~A ~~ n <-> P~m ~~ n))
7	5, 6	mpan 693	. . . . . . 7 |- (P~A ~~ P~m -> (P~A ~~ n <-> P~m ~~ n))
8	4, 7	syl 10	. . . . . 6 |- (A ~~ m -> (P~A ~~ n <-> P~m ~~ n))
9	8	rexbidv 1656	. . . . 5 |- (A ~~ m -> (E.n e. om P~A ~~ n <-> E.n e. om P~m ~~ n))
10		pweq 2393	. . . . . . . 8 |- (m = (/) -> P~m = P~(/))
11	10	breq1d 2619	. . . . . . 7 |- (m = (/) -> (P~m ~~ n <-> P~(/) ~~ n))
12	11	rexbidv 1656	. . . . . 6 |- (m = (/) -> (E.n e. om P~m ~~ n <-> E.n e. om P~(/) ~~ n))
13		pweq 2393	. . . . . . . 8 |- (m = k -> P~m = P~k)
14	13	breq1d 2619	. . . . . . 7 |- (m = k -> (P~m ~~ n <-> P~k ~~ n))
15	14	rexbidv 1656	. . . . . 6 |- (m = k -> (E.n e. om P~m ~~ n <-> E.n e. om P~k ~~ n))
16		df-suc 2944	. . . . . . . . . 10 |- suc k = (k u. {k})
17	16	eqeq2i 1477	. . . . . . . . 9 |- (m = suc k <-> m = (k u. {k}))
18		pweq 2393	. . . . . . . . 9 |- (m = (k u. {k}) -> P~m = P~(k u. {k}))
19	17, 18	sylbi 199	. . . . . . . 8 |- (m = suc k -> P~m = P~(k u. {k}))
20	19	breq1d 2619	. . . . . . 7 |- (m = suc k -> (P~m ~~ n <-> P~(k u. {k}) ~~ n))
21	20	rexbidv 1656	. . . . . 6 |- (m = suc k -> (E.n e. om P~m ~~ n <-> E.n e. om P~(k u. {k}) ~~ n))
22		1onn 4237	. . . . . . 7 |- 1o e. om
23		pw0 2459	. . . . . . . . 9 |- P~(/) = {(/)}
24		df1o2 4124	. . . . . . . . 9 |- 1o = {(/)}
25	23, 24	eqtr4 1490	. . . . . . . 8 |- P~(/) = 1o
26	22	elisseti 1809	. . . . . . . . 9 |- 1o e. V
27	26	enref 4372	. . . . . . . 8 |- 1o ~~ 1o
28	25, 27	eqbrtr 2624	. . . . . . 7 |- P~(/) ~~ 1o
29		breq2 2613	. . . . . . . 8 |- (n = 1o -> (P~(/) ~~ n <-> P~(/) ~~ 1o))
30	29	rcla4ev 1868	. . . . . . 7 |- ((1o e. om /\ P~(/) ~~ 1o) -> E.n e. om P~(/) ~~ n)
31	22, 28, 30	mp2an 695	. . . . . 6 |- E.n e. om P~(/) ~~ n
32		eqid 1468	. . . . . . . 8 |- {<.c, y>. | (c e. P~k /\ y = (c u. {k}))} = {<.c, y>. | (c e. P~k /\ y = (c u. {k}))}
33	32	pwfilem 4544	. . . . . . 7 |- (E.n e. om P~k ~~ n -> E.n e. om P~(k u. {k}) ~~ n)
34	33	a1i 8	. . . . . 6 |- (k e. om -> (E.n e. om P~k ~~ n -> E.n e. om P~(k u. {k}) ~~ n))
35	12, 15, 21, 31, 34	finds1 3149	. . . . 5 |- (m e. om -> E.n e. om P~m ~~ n)
36	9, 35	syl5cbir 211	. . . 4 |- (m e. om -> (A ~~ m -> E.n e. om P~A ~~ n))
37	36	r19.23aiv 1735	. . 3 |- (E.m e. om A ~~ m -> E.n e. om P~A ~~ n)
38	2, 37	sylbi 199	. 2 |- (E.n e. om A ~~ n -> E.n e. om P~A ~~ n)
39		relen 4354	. . . . . . . . 9 |- Rel ~~
40	39	brrelexi 3198	. . . . . . . 8 |- (P~A ~~ n -> P~A e. V)
41		pwexb 2898	. . . . . . . 8 |- (A e. V <-> P~A e. V)
42	40, 41	sylibr 200	. . . . . . 7 |- (P~A ~~ n -> A e. V)
43		canth2g 4466	. . . . . . 7 |- (A e. V -> A ~< P~A)
44	42, 43	syl 10	. . . . . 6 |- (P~A ~~ n -> A ~< P~A)
45	44	adantl 388	. . . . 5 |- ((n e. om /\ P~A ~~ n) -> A ~< P~A)
46	45	r19.23aiva 1736	. . . 4 |- (E.n e. om P~A ~~ n -> A ~< P~A)
47		sdomdom 4367	. . . 4 |- (A ~< P~A -> A ~<_ P~A)
48	46, 47	syl 10	. . 3 |- (E.n e. om P~A ~~ n -> A ~<_ P~A)
49		domfi 4516	. . 3 |- ((E.n e. om P~A ~~ n /\ A ~<_ P~A) -> E.n e. om A ~~ n)
50	48, 49	mpdan 702	. 2 |- (E.n e. om P~A ~~ n -> E.n e. om A ~~ n)
51	38, 50	impbi 157	1 |- (E.n e. om A ~~ n <-> E.n e. om P~A ~~ n)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638  Vcvv 1802   u. cun 2035  (/)c0 2270  P~cpw 2391  {csn 2399   class class class wbr 2609  {copab 2656  suc csuc 2940  omcom 3121  1oc1o 4112   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  dominf 4876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-2o 4118  df-oadd 4119  df-er 4245  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353

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