Theorem brdom5 5964

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 5963	. . 3 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
4		alral 2153	. . . . 5 |- (A.xE*y xfy -> A.x e. B E*y xfy)
5	4	anim1i 361	. . . 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	eximi 1387	. . 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 216	. 2 |- (A ~<_ B -> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
8		inss2 2813	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) C_ (B X. A)
9		dmss 4156	. . . . . . . . . . . . . 14 |- ((f i^i (B X. A)) C_ (B X. A) -> dom ( f i^i (B X. A)) C_ dom ( B X. A))
10	8, 9	ax-mp 7	. . . . . . . . . . . . 13 |- dom ( f i^i (B X. A)) C_ dom ( B X. A)
11		dmxpss 4343	. . . . . . . . . . . . 13 |- dom ( B X. A) C_ B
12	10, 11	sstri 2626	. . . . . . . . . . . 12 |- dom ( f i^i (B X. A)) C_ B
13	12	sseli 2617	. . . . . . . . . . 11 |- (x e. dom ( f i^i (B X. A)) -> x e. B)
14		inss1 2812	. . . . . . . . . . . . 13 |- (f i^i (B X. A)) C_ f
15	14	ssbri 3379	. . . . . . . . . . . 12 |- (x(f i^i (B X. A))y -> xfy)
16	15	immoi 1814	. . . . . . . . . . 11 |- (E*y xfy -> E*y x(f i^i (B X. A))y)
17	13, 16	imim12i 21	. . . . . . . . . 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	ralimi2 2165	. . . . . . . . 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 4088	. . . . . . . . . 10 |- Rel (B X. A)
20		relin2 4099	. . . . . . . . . 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 316	. . . . . . . 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		dffun7 4447	. . . . . . . 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 217	. . . . . . 7 |- (A.x e. B E*y xfy -> Fun (f i^i (B X. A)))
25		funfn 4451	. . . . . . 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 215	. . . . . 6 |- (A.x e. B E*y xfy -> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
27		rninxp 4355	. . . . . . 7 |- (ran ( f i^i (B X. A)) = A <-> A.x e. A E.y e. B yfx)
28	27	biimpri 169	. . . . . 6 |- (A.x e. A E.y e. B yfx -> ran ( f i^i (B X. A)) = A)
29	26, 28	anim12i 360	. . . . 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 4012	. . . . 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 217	. . . 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 2295	. . . . . . 7 |- f e. _V
33	32	inex1 3452	. . . . . 6 |- (f i^i (B X. A)) e. _V
34	33	dmex 4208	. . . . 5 |- dom ( f i^i (B X. A)) e. _V
35	34	fodom 5960	. . . 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 5468	. . . . . 6 |- (B e. _V -> (dom ( f i^i (B X. A)) C_ B -> dom ( f i^i (B X. A)) ~<_ B))
37	2, 12, 36	mp2 54	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
38		domtr 5474	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
39	37, 38	mpan2 760	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
40	31, 35, 39	3syl 24	. . 3 |- ((A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
41	40	19.23aiv 1674	. 2 |- (E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
42	7, 41	impbii 174	1 |- (A ~<_ B <-> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))

Syntax hints:   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E*wmo 1772  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593   class class class wbr 3338   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996   ~<_ cdom 5424

This theorem is referenced by:  brdom6disj 5967

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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