Theorem brdom4 4727

Description: An equivalence to a dominance relation.

Hypotheses

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

Assertion

Ref	Expression
brdom4	|- (A ~<_ B <-> E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))

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

Proof of Theorem brdom4

Step	Hyp	Ref	Expression
1		brdom4.1	. . . 4 |- A e. V
2		brdom4.2	. . . 4 |- B e. V
3	1, 2	brdom3 4725	. . 3 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
4		moan 1399	. . . . . . 7 |- (E*y xfy -> E*y(y e. A /\ xfy))
5	4	19.20i 968	. . . . . 6 |- (A.xE*y xfy -> A.xE*y(y e. A /\ xfy))
6		alral 1668	. . . . . 6 |- (A.xE*y(y e. A /\ xfy) -> A.x e. B E*y(y e. A /\ xfy))
7	5, 6	syl 10	. . . . 5 |- (A.xE*y xfy -> A.x e. B E*y(y e. A /\ xfy))
8	7	anim1i 334	. . . 4 |- ((A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> (A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))
9	8	19.22i 1016	. . 3 |- (E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))
10	3, 9	sylbi 199	. 2 |- (A ~<_ B -> E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))
11		inss2 2202	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) (_ (B X. A)
12		dmss 3267	. . . . . . . . . . . . . 14 |- ((f i^i (B X. A)) (_ (B X. A) -> dom ( f i^i (B X. A)) (_ dom ( B X. A))
13	11, 12	ax-mp 7	. . . . . . . . . . . . 13 |- dom ( f i^i (B X. A)) (_ dom ( B X. A)
14		dmxpss 3422	. . . . . . . . . . . . 13 |- dom ( B X. A) (_ B
15	13, 14	sstri 2044	. . . . . . . . . . . 12 |- dom ( f i^i (B X. A)) (_ B
16	15	sseli 2036	. . . . . . . . . . 11 |- (x e. dom ( f i^i (B X. A)) -> x e. B)
17		rnss 3301	. . . . . . . . . . . . . . . 16 |- ((f i^i (B X. A)) (_ (B X. A) -> ran ( f i^i (B X. A)) (_ ran ( B X. A))
18	11, 17	ax-mp 7	. . . . . . . . . . . . . . 15 |- ran ( f i^i (B X. A)) (_ ran ( B X. A)
19		rnxpss 3423	. . . . . . . . . . . . . . 15 |- ran ( B X. A) (_ A
20	18, 19	sstri 2044	. . . . . . . . . . . . . 14 |- ran ( f i^i (B X. A)) (_ A
21	20	sseli 2036	. . . . . . . . . . . . 13 |- (y e. ran ( f i^i (B X. A)) -> y e. A)
22		inss1 2201	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) (_ f
23	22	ssbri 2625	. . . . . . . . . . . . 13 |- (x(f i^i (B X. A))y -> xfy)
24	21, 23	anim12i 333	. . . . . . . . . . . 12 |- ((y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y) -> (y e. A /\ xfy))
25	24	immoi 1395	. . . . . . . . . . 11 |- (E*y(y e. A /\ xfy) -> E*y(y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y))
26	16, 25	imim12i 18	. . . . . . . . . 10 |- ((x e. B -> E*y(y e. A /\ xfy)) -> (x e. dom ( f i^i (B X. A)) -> E*y(y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y)))
27	26	r19.20i2 1679	. . . . . . . . 9 |- (A.x e. B E*y(y e. A /\ xfy) -> A.x e. dom ( f i^i (B X. A))E*y(y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y))
28		relxp 3217	. . . . . . . . . 10 |- Rel (B X. A)
29		relin2 3225	. . . . . . . . . 10 |- (Rel (B X. A) -> Rel (f i^i (B X. A)))
30	28, 29	ax-mp 7	. . . . . . . . 9 |- Rel (f i^i (B X. A))
31	27, 30	jctil 292	. . . . . . . 8 |- (A.x e. B E*y(y e. A /\ xfy) -> (Rel (f i^i (B X. A)) /\ A.x e. dom ( f i^i (B X. A))E*y(y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y)))
32		dffun8 3482	. . . . . . . 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(y e. ran ( f i^i (B X. A)) /\ x(f i^i (B X. A))y)))
33	31, 32	sylibr 200	. . . . . . 7 |- (A.x e. B E*y(y e. A /\ xfy) -> Fun (f i^i (B X. A)))
34		funfn 3483	. . . . . . 7 |- (Fun (f i^i (B X. A)) <-> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
35	33, 34	sylib 198	. . . . . 6 |- (A.x e. B E*y(y e. A /\ xfy) -> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
36		rninxp 3428	. . . . . . 7 |- (ran ( f i^i (B X. A)) = A <-> A.x e. A E.y e. B yfx)
37	36	biimpr 152	. . . . . 6 |- (A.x e. A E.y e. B yfx -> ran ( f i^i (B X. A)) = A)
38	35, 37	anim12i 333	. . . . 5 |- ((A.x e. B E*y(y e. A /\ 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))
39		df-fo 3159	. . . . 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))
40	38, 39	sylibr 200	. . . 4 |- ((A.x e. B E*y(y e. A /\ 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)
41		visset 1788	. . . . . . 7 |- f e. V
42	41	inex1 2684	. . . . . 6 |- (f i^i (B X. A)) e. V
43		dmexg 3289	. . . . . 6 |- ((f i^i (B X. A)) e. V -> dom ( f i^i (B X. A)) e. V)
44	42, 43	ax-mp 7	. . . . 5 |- dom ( f i^i (B X. A)) e. V
45	44	fodom 4722	. . . 4 |- ((f i^i (B X. A)):dom ( f i^i (B X. A))-onto->A -> A ~<_ dom ( f i^i (B X. A)))
46		ssdom2g 4344	. . . . . 6 |- (B e. V -> (dom ( f i^i (B X. A)) (_ B -> dom ( f i^i (B X. A)) ~<_ B))
47	2, 15, 46	mp2 43	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
48		domtr 4350	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
49	47, 48	mpan2 693	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
50	40, 45, 49	3syl 20	. . 3 |- ((A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
51	50	19.23aiv 1277	. 2 |- (E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
52	10, 51	impbi 157	1 |- (A ~<_ B <-> E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  E*wmo 1358  A.wral 1621  E.wrex 1622  Vcvv 1786   i^i cin 2017   (_ wss 2018   class class class wbr 2587   X. cxp 3131  dom cdm 3133  ran crn 3134  Rel wrel 3138  Fun wfun 3139   Fn wfn 3140  -onto->wfo 3143   ~<_ cdom 4303

This theorem is referenced by:  brdom7disj 4728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-fv 3161  df-er 4199  df-en 4305  df-dom 4306  df-sdom 4307

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