Theorem brdom5 4802

Description: An equivalence to a dominance relation.

Hypotheses

Ref	Expression
brdom4.1	|- A e. V
brdom4.2	|- B e. V

Assertion

Ref	Expression
brdom5	|- (A ~<_ B <-> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))

Distinct variable groups:   x,f,y,A   B,f,x,y

Proof of Theorem brdom5

Step	Hyp	Ref	Expression
1		brdom4.1	. . . 4 |- A e. V
2		brdom4.2	. . . 4 |- B e. V
3	1, 2	brdom3 4801	. . 3 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
4		alral 1692	. . . . 5 |- (A.xE*y xfy -> A.x e. B E*y xfy)
5	4	anim1i 334	. . . 4 |- ((A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> (A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
6	5	19.22i 1040	. . 3 |- (E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
7	3, 6	sylbi 199	. 2 |- (A ~<_ B -> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
8		inss2 2231	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) (_ (B X. A)
9		dmss 3310	. . . . . . . . . . . . . 14 |- ((f i^i (B X. A)) (_ (B X. A) -> dom ( f i^i (B X. A)) (_ dom ( B X. A))
10	8, 9	ax-mp 7	. . . . . . . . . . . . 13 |- dom ( f i^i (B X. A)) (_ dom ( B X. A)
11		dmxpss 3473	. . . . . . . . . . . . 13 |- dom ( B X. A) (_ B
12	10, 11	sstri 2073	. . . . . . . . . . . 12 |- dom ( f i^i (B X. A)) (_ B
13	12	sseli 2065	. . . . . . . . . . 11 |- (x e. dom ( f i^i (B X. A)) -> x e. B)
14		inss1 2230	. . . . . . . . . . . . 13 |- (f i^i (B X. A)) (_ f
15	14	ssbri 2657	. . . . . . . . . . . 12 |- (x(f i^i (B X. A))y -> xfy)
16	15	immoi 1418	. . . . . . . . . . 11 |- (E*y xfy -> E*y x(f i^i (B X. A))y)
17	13, 16	imim12i 18	. . . . . . . . . 10 |- ((x e. B -> E*y xfy) -> (x e. dom ( f i^i (B X. A)) -> E*y x(f i^i (B X. A))y))
18	17	r19.20i2 1703	. . . . . . . . 9 |- (A.x e. B E*y xfy -> A.x e. dom ( f i^i (B X. A))E*y x(f i^i (B X. A))y)
19		relxp 3255	. . . . . . . . . 10 |- Rel (B X. A)
20		relin2 3263	. . . . . . . . . 10 |- (Rel (B X. A) -> Rel (f i^i (B X. A)))
21	19, 20	ax-mp 7	. . . . . . . . 9 |- Rel (f i^i (B X. A))
22	18, 21	jctil 292	. . . . . . . 8 |- (A.x e. B E*y xfy -> (Rel (f i^i (B X. A)) /\ A.x e. dom ( f i^i (B X. A))E*y x(f i^i (B X. A))y))
23		dffun6 3539	. . . . . . . 8 |- (Fun (f i^i (B X. A)) <-> (Rel (f i^i (B X. A)) /\ A.x e. dom ( f i^i (B X. A))E*y x(f i^i (B X. A))y))
24	22, 23	sylibr 200	. . . . . . 7 |- (A.x e. B E*y xfy -> Fun (f i^i (B X. A)))
25		funfn 3542	. . . . . . 7 |- (Fun (f i^i (B X. A)) <-> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
26	24, 25	sylib 198	. . . . . 6 |- (A.x e. B E*y xfy -> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
27		rninxp 3482	. . . . . . 7 |- (ran ( f i^i (B X. A)) = A <-> A.x e. A E.y e. B yfx)
28	27	biimpr 152	. . . . . 6 |- (A.x e. A E.y e. B yfx -> ran ( f i^i (B X. A)) = A)
29	26, 28	anim12i 333	. . . . 5 |- ((A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> ((f i^i (B X. A)) Fn dom ( f i^i (B X. A)) /\ ran ( f i^i (B X. A)) = A))
30		df-fo 3196	. . . . 5 |- ((f i^i (B X. A)):dom ( f i^i (B X. A))-onto->A <-> ((f i^i (B X. A)) Fn dom ( f i^i (B X. A)) /\ ran ( f i^i (B X. A)) = A))
31	29, 30	sylibr 200	. . . 4 |- ((A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> (f i^i (B X. A)):dom ( f i^i (B X. A))-onto->A)
32		visset 1813	. . . . . . 7 |- f e. V
33	32	inex1 2716	. . . . . 6 |- (f i^i (B X. A)) e. V
34	33	dmex 3360	. . . . 5 |- dom ( f i^i (B X. A)) e. V
35	34	fodom 4798	. . . 4 |- ((f i^i (B X. A)):dom ( f i^i (B X. A))-onto->A -> A ~<_ dom ( f i^i (B X. A)))
36		ssdom2g 4409	. . . . . 6 |- (B e. V -> (dom ( f i^i (B X. A)) (_ B -> dom ( f i^i (B X. A)) ~<_ B))
37	2, 12, 36	mp2 43	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
38		domtr 4415	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
39	37, 38	mpan2 696	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
40	31, 35, 39	3syl 20	. . 3 |- ((A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
41	40	19.23aiv 1295	. 2 |- (E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
42	7, 41	impbi 157	1 |- (A ~<_ B <-> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  E*wmo 1381  A.wral 1645  E.wrex 1646  Vcvv 1811   i^i cin 2046   (_ wss 2047   class class class wbr 2619   X. cxp 3168  dom cdm 3170  ran crn 3171  Rel wrel 3175  Fun wfun 3176   Fn wfn 3177  -onto->wfo 3180   ~<_ cdom 4365

This theorem is referenced by:  brdom6disj 4805

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370

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