Theorem brdom5 4785

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 4784	. . 3 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
4		alral 1690	. . . . 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 1039	. . 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 2228	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) (_ (B X. A)
9		dmss 3306	. . . . . . . . . . . . . 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 3469	. . . . . . . . . . . . 13 |- dom ( B X. A) (_ B
12	10, 11	sstri 2070	. . . . . . . . . . . 12 |- dom ( f i^i (B X. A)) (_ B
13	12	sseli 2062	. . . . . . . . . . 11 |- (x e. dom ( f i^i (B X. A)) -> x e. B)
14		inss1 2227	. . . . . . . . . . . . 13 |- (f i^i (B X. A)) (_ f
15	14	ssbri 2653	. . . . . . . . . . . 12 |- (x(f i^i (B X. A))y -> xfy)
16	15	immoi 1417	. . . . . . . . . . 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 1701	. . . . . . . . 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 3251	. . . . . . . . . 10 |- Rel (B X. A)
20		relin2 3259	. . . . . . . . . 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 3535	. . . . . . . 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 3538	. . . . . . 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 3478	. . . . . . 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 3192	. . . . 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 1810	. . . . . . 7 |- f e. V
33	32	inex1 2712	. . . . . 6 |- (f i^i (B X. A)) e. V
34	33	dmex 3356	. . . . 5 |- dom ( f i^i (B X. A)) e. V
35	34	fodom 4781	. . . 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 4399	. . . . . 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 4405	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
39	37, 38	mpan2 695	. . . 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 1294	. 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 953   = wceq 955   e. wcel 957  E.wex 979  E*wmo 1380  A.wral 1643  E.wrex 1644  Vcvv 1808   i^i cin 2043   (_ wss 2044   class class class wbr 2615   X. cxp 3164  dom cdm 3166  ran crn 3167  Rel wrel 3171  Fun wfun 3172   Fn wfn 3173  -onto->wfo 3176   ~<_ cdom 4358

This theorem is referenced by:  brdom6disj 4788

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-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  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-er 4254  df-en 4360  df-dom 4361  df-sdom 4362

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