Theorem brdom3 4725

Description: Equivalence to a dominance relation.

Hypotheses

Ref	Expression
brdom3.1	|- A e. V
brdom3.2	|- B e. V

Assertion

Ref	Expression
brdom3	|- (A ~<_ B <-> E.f(A.xE*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 brdom3

Step	Hyp	Ref	Expression
1		brdom3.2	. . . . . . 7 |- B e. V
2		fodomr 4417	. . . . . . 7 |- ((B e. V /\ (/) ~< A /\ A ~<_ B) -> E.f f:B-onto->A)
3	1, 2	mp3an1 899	. . . . . 6 |- (((/) ~< A /\ A ~<_ B) -> E.f f:B-onto->A)
4		brdom3.1	. . . . . . . 8 |- A e. V
5	4	0sdom 4401	. . . . . . 7 |- ((/) ~< A <-> A =/= (/))
6		df-ne 1563	. . . . . . 7 |- (A =/= (/) <-> -. A = (/))
7	5, 6	bitr2 174	. . . . . 6 |- (-. A = (/) <-> (/) ~< A)
8	3, 7	sylanb 449	. . . . 5 |- ((-. A = (/) /\ A ~<_ B) -> E.f f:B-onto->A)
9	8	ancoms 436	. . . 4 |- ((A ~<_ B /\ -. A = (/)) -> E.f f:B-onto->A)
10		pm5.6 685	. . . 4 |- (((A ~<_ B /\ -. A = (/)) -> E.f f:B-onto->A) <-> (A ~<_ B -> (A = (/) \/ E.f f:B-onto->A)))
11	9, 10	mpbi 189	. . 3 |- (A ~<_ B -> (A = (/) \/ E.f f:B-onto->A))
12		rzal 2326	. . . . . 6 |- (A = (/) -> A.x e. A E.y e. B y(/)x)
13		noel 2255	. . . . . . . . . 10 |- -. <.x, y>. e. (/)
14		df-br 2588	. . . . . . . . . 10 |- (x(/)y <-> <.x, y>. e. (/))
15	13, 14	mtbir 192	. . . . . . . . 9 |- -. x(/)y
16	15	nex 1077	. . . . . . . 8 |- -. E.y x(/)y
17		exmo 1393	. . . . . . . . 9 |- (E.y x(/)y \/ E*y x(/)y)
18	17	ori 230	. . . . . . . 8 |- (-. E.y x(/)y -> E*y x(/)y)
19	16, 18	ax-mp 7	. . . . . . 7 |- E*y x(/)y
20	19	ax-gen 955	. . . . . 6 |- A.xE*y x(/)y
21	12, 20	jctil 292	. . . . 5 |- (A = (/) -> (A.xE*y x(/)y /\ A.x e. A E.y e. B y(/)x))
22		0ex 2679	. . . . . 6 |- (/) e. V
23		ax-17 1190	. . . . . . . . 9 |- (f = (/) -> A.y f = (/))
24		breq 2589	. . . . . . . . 9 |- (f = (/) -> (xfy <-> x(/)y))
25	23, 24	mobid 1381	. . . . . . . 8 |- (f = (/) -> (E*y xfy <-> E*y x(/)y))
26	25	albidv 1260	. . . . . . 7 |- (f = (/) -> (A.xE*y xfy <-> A.xE*y x(/)y))
27		breq 2589	. . . . . . . . 9 |- (f = (/) -> (yfx <-> y(/)x))
28	27	rexbidv 1640	. . . . . . . 8 |- (f = (/) -> (E.y e. B yfx <-> E.y e. B y(/)x))
29	28	ralbidv 1639	. . . . . . 7 |- (f = (/) -> (A.x e. A E.y e. B yfx <-> A.x e. A E.y e. B y(/)x))
30	26, 29	anbi12d 626	. . . . . 6 |- (f = (/) -> ((A.xE*y xfy /\ A.x e. A E.y e. B yfx) <-> (A.xE*y x(/)y /\ A.x e. A E.y e. B y(/)x)))
31	22, 30	cla4ev 1842	. . . . 5 |- ((A.xE*y x(/)y /\ A.x e. A E.y e. B y(/)x) -> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
32	21, 31	syl 10	. . . 4 |- (A = (/) -> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
33		fofun 3612	. . . . . . 7 |- (f:B-onto->A -> Fun f)
34		dffunmo 3472	. . . . . . . 8 |- (Fun f <-> (Rel f /\ A.xE*y xfy))
35	34	pm3.27bi 326	. . . . . . 7 |- (Fun f -> A.xE*y xfy)
36	33, 35	syl 10	. . . . . 6 |- (f:B-onto->A -> A.xE*y xfy)
37		dffo4 3759	. . . . . . 7 |- (f:B-onto->A <-> (f:B-->A /\ A.x e. A E.y e. B yfx))
38	37	pm3.27bi 326	. . . . . 6 |- (f:B-onto->A -> A.x e. A E.y e. B yfx)
39	36, 38	jca 288	. . . . 5 |- (f:B-onto->A -> (A.xE*y xfy /\ A.x e. A E.y e. B yfx))
40	39	19.22i 1016	. . . 4 |- (E.f f:B-onto->A -> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
41	32, 40	jaoi 341	. . 3 |- ((A = (/) \/ E.f f:B-onto->A) -> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
42	11, 41	syl 10	. 2 |- (A ~<_ B -> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
43		inss1 2201	. . . . . . . . . . 11 |- (f i^i (B X. A)) (_ f
44	43	ssbri 2625	. . . . . . . . . 10 |- (x(f i^i (B X. A))y -> xfy)
45	44	immoi 1395	. . . . . . . . 9 |- (E*y xfy -> E*y x(f i^i (B X. A))y)
46	45	19.20i 968	. . . . . . . 8 |- (A.xE*y xfy -> A.xE*y x(f i^i (B X. A))y)
47		dffunmo 3472	. . . . . . . . 9 |- (Fun (f i^i (B X. A)) <-> (Rel (f i^i (B X. A)) /\ A.xE*y x(f i^i (B X. A))y))
48		relxp 3217	. . . . . . . . . 10 |- Rel (B X. A)
49		relin2 3225	. . . . . . . . . 10 |- (Rel (B X. A) -> Rel (f i^i (B X. A)))
50	48, 49	ax-mp 7	. . . . . . . . 9 |- Rel (f i^i (B X. A))
51	47, 50	mpbiran 725	. . . . . . . 8 |- (Fun (f i^i (B X. A)) <-> A.xE*y x(f i^i (B X. A))y)
52	46, 51	sylibr 200	. . . . . . 7 |- (A.xE*y xfy -> Fun (f i^i (B X. A)))
53		funfn 3483	. . . . . . 7 |- (Fun (f i^i (B X. A)) <-> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
54	52, 53	sylib 198	. . . . . 6 |- (A.xE*y xfy -> (f i^i (B X. A)) Fn dom ( f i^i (B X. A)))
55		rninxp 3428	. . . . . . 7 |- (ran ( f i^i (B X. A)) = A <-> A.x e. A E.y e. B yfx)
56	55	biimpr 152	. . . . . 6 |- (A.x e. A E.y e. B yfx -> ran ( f i^i (B X. A)) = A)
57	54, 56	anim12i 333	. . . . 5 |- ((A.xE*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))
58		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))
59	57, 58	sylibr 200	. . . 4 |- ((A.xE*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)
60		visset 1788	. . . . . . 7 |- f e. V
61	60	inex1 2684	. . . . . 6 |- (f i^i (B X. A)) e. V
62		dmexg 3289	. . . . . 6 |- ((f i^i (B X. A)) e. V -> dom ( f i^i (B X. A)) e. V)
63	61, 62	ax-mp 7	. . . . 5 |- dom ( f i^i (B X. A)) e. V
64	63	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)))
65		inss2 2202	. . . . . . . 8 |- (f i^i (B X. A)) (_ (B X. A)
66		dmss 3267	. . . . . . . 8 |- ((f i^i (B X. A)) (_ (B X. A) -> dom ( f i^i (B X. A)) (_ dom ( B X. A))
67	65, 66	ax-mp 7	. . . . . . 7 |- dom ( f i^i (B X. A)) (_ dom ( B X. A)
68		dmxpss 3422	. . . . . . 7 |- dom ( B X. A) (_ B
69	67, 68	sstri 2044	. . . . . 6 |- dom ( f i^i (B X. A)) (_ B
70		ssdom2g 4344	. . . . . 6 |- (B e. V -> (dom ( f i^i (B X. A)) (_ B -> dom ( f i^i (B X. A)) ~<_ B))
71	1, 69, 70	mp2 43	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
72		domtr 4350	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
73	71, 72	mpan2 693	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
74	59, 64, 73	3syl 20	. . 3 |- ((A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
75	74	19.23aiv 1277	. 2 |- (E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
76	42, 75	impbi 157	1 |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  E*wmo 1358   =/= wne 1561  A.wral 1621  E.wrex 1622  Vcvv 1786   i^i cin 2017   (_ wss 2018  (/)c0 2251  <.cop 2382   class class class wbr 2587   X. cxp 3131  dom cdm 3133  ran crn 3134  Rel wrel 3138  Fun wfun 3139   Fn wfn 3140  -->wf 3141  -onto->wfo 3143   ~<_ cdom 4303   ~< csdm 4304

This theorem is referenced by:  brdom5 4726  brdom4 4727

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