Theorem brdom5 6378

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 6377	. . 3 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
4		alral 2404	. . . . 5 |- (A.xE*y xfy -> A.x e. B E*y xfy)
5	4	anim1i 538	. . . 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 1676	. . 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 225	. 2 |- (A ~<_ B -> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
8		inss2 3027	. . . . . . . . . . . . . 14 |- (f i^i (B X. A)) C_ (B X. A)
9		dmss 4282	. . . . . . . . . . . . . 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 4450	. . . . . . . . . . . . 13 |- dom ( B X. A) C_ B
12	10, 11	sstri 2856	. . . . . . . . . . . 12 |- dom ( f i^i (B X. A)) C_ B
13	12	sseli 2848	. . . . . . . . . . 11 |- (x e. dom ( f i^i (B X. A)) -> x e. B)
14		inss1 3026	. . . . . . . . . . . . 13 |- (f i^i (B X. A)) C_ f
15	14	ssbri 3553	. . . . . . . . . . . 12 |- (x(f i^i (B X. A))y -> xfy)
16	15	immoi 2079	. . . . . . . . . . 11 |- (E*y xfy -> E*y x(f i^i (B X. A))y)
17	13, 16	imim12i 38	. . . . . . . . . 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 2415	. . . . . . . . 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 4219	. . . . . . . . . 10 |- Rel (B X. A)
20		relin2 4229	. . . . . . . . . 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 501	. . . . . . . 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 4550	. . . . . . . 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 243	. . . . . . 7 |- (A.x e. B E*y xfy -> Fun (f i^i (B X. A)))
25		funfn 4553	. . . . . . 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 242	. . . . . 6 |- (A.x e. B E*y xfy -> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
27		rninxp 4461	. . . . . . 7 |- (ran ( f i^i (B X. A)) = A <-> A.x e. A E.y e. B yfx)
28	27	biimpri 230	. . . . . 6 |- (A.x e. A E.y e. B yfx -> ran ( f i^i (B X. A)) = A)
29	26, 28	anim12i 536	. . . . 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 4145	. . . . 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 243	. . . 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 2541	. . . . . . 7 |- f e. _V
33	32	inex1 3619	. . . . . 6 |- (f i^i (B X. A)) e. _V
34	33	dmex 4330	. . . . 5 |- dom ( f i^i (B X. A)) e. _V
35	34	fodom 6374	. . . 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 5629	. . . . . 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 95	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
38		domtr 5635	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
39	37, 38	mpan2 679	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
40	31, 35, 39	3syl 41	. . 3 |- ((A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
41	40	19.23aiv 1943	. 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 223	1 |- (A ~<_ B <-> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))

Syntax hints:   <-> wb 219   /\ wa 337  A.wal 1584   = wceq 1586   e. wcel 1588  E.wex 1615  E*wmo 2038  A.wral 2355  E.wrex 2356  _Vcvv 2538   i^i cin 2826   C_ wss 2827   class class class wbr 3507   X. cxp 4117  dom cdm 4119  ran crn 4120  Rel wrel 4124  Fun wfun 4125   Fn wfn 4126  -onto->wfo 4129   ~<_ cdom 5585

This theorem is referenced by:  brdom6disj 6381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-13 1599  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-rep 3596  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-pr 3687  ax-un 3929  ax-ac 6314

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-3an 1104  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-reu 2361  df-rab 2362  df-v 2540  df-dif 2830  df-un 2832  df-in 2834  df-ss 2836  df-nul 3083  df-pw 3229  df-sn 3242  df-pr 3243  df-op 3246  df-uni 3367  df-br 3508  df-opab 3566  df-id 3747  df-xp 4133  df-rel 4134  df-cnv 4135  df-co 4136  df-dm 4137  df-rn 4138  df-res 4139  df-ima 4140  df-fun 4141  df-fn 4142  df-f 4143  df-f1 4144  df-fo 4145  df-f1o 4146  df-fv 4147  df-er 5479  df-en 5588  df-dom 5589  df-sdom 5590

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