Theorem brdom3 4788

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 4476	. . . . . . 7 |- ((B e. V /\ (/) ~< A /\ A ~<_ B) -> E.f f:B-onto->A)
3	1, 2	mp3an1 902	. . . . . 6 |- (((/) ~< A /\ A ~<_ B) -> E.f f:B-onto->A)
4		brdom3.1	. . . . . . . 8 |- A e. V
5	4	0sdom 4460	. . . . . . 7 |- ((/) ~< A <-> A =/= (/))
6		df-ne 1586	. . . . . . 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		pm5.6 687	. . . 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 2353	. . . . . 6 |- (A = (/) -> A.x e. A E.y e. B y(/)x)
13		noel 2282	. . . . . . . . . 10 |- -. <.x, y>. e. (/)
14		df-br 2617	. . . . . . . . . 10 |- (x(/)y <-> <.x, y>. e. (/))
15	13, 14	mtbir 192	. . . . . . . . 9 |- -. x(/)y
16	15	nex 1100	. . . . . . . 8 |- -. E.y x(/)y
17		exmo 1416	. . . . . . . . 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 962	. . . . . 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 2708	. . . . . 6 |- (/) e. V
23		ax-17 970	. . . . . . . . 9 |- (f = (/) -> A.y f = (/))
24		breq 2618	. . . . . . . . 9 |- (f = (/) -> (xfy <-> x(/)y))
25	23, 24	mobid 1404	. . . . . . . 8 |- (f = (/) -> (E*y xfy <-> E*y x(/)y))
26	25	albidv 1278	. . . . . . 7 |- (f = (/) -> (A.xE*y xfy <-> A.xE*y x(/)y))
27		breq 2618	. . . . . . . . 9 |- (f = (/) -> (yfx <-> y(/)x))
28	27	rexbidv 1663	. . . . . . . 8 |- (f = (/) -> (E.y e. B yfx <-> E.y e. B y(/)x))
29	28	ralbidv 1662	. . . . . . 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 627	. . . . . 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 1867	. . . . 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 3670	. . . . . . 7 |- (f:B-onto->A -> Fun f)
34		dffunmo 3528	. . . . . . . 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 3817	. . . . . . 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 1039	. . . 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 2228	. . . . . . . . . . 11 |- (f i^i (B X. A)) (_ f
44	43	ssbri 2654	. . . . . . . . . 10 |- (x(f i^i (B X. A))y -> xfy)
45	44	immoi 1418	. . . . . . . . 9 |- (E*y xfy -> E*y x(f i^i (B X. A))y)
46	45	19.20i 991	. . . . . . . 8 |- (A.xE*y xfy -> A.xE*y x(f i^i (B X. A))y)
47		dffunmo 3528	. . . . . . . . 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 3252	. . . . . . . . . 10 |- Rel (B X. A)
49		relin2 3260	. . . . . . . . . 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 727	. . . . . . . 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 3539	. . . . . . 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 3479	. . . . . . 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 3193	. . . . 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 1811	. . . . . . 7 |- f e. V
61	60	inex1 2713	. . . . . 6 |- (f i^i (B X. A)) e. V
62	61	dmex 3357	. . . . 5 |- dom ( f i^i (B X. A)) e. V
63	62	fodom 4785	. . . 4 |- ((f i^i (B X. A)):dom ( f i^i (B X. A))-onto->A -> A ~<_ dom ( f i^i (B X. A)))
64		inss2 2229	. . . . . . . 8 |- (f i^i (B X. A)) (_ (B X. A)
65		dmss 3307	. . . . . . . 8 |- ((f i^i (B X. A)) (_ (B X. A) -> dom ( f i^i (B X. A)) (_ dom ( B X. A))
66	64, 65	ax-mp 7	. . . . . . 7 |- dom ( f i^i (B X. A)) (_ dom ( B X. A)
67		dmxpss 3470	. . . . . . 7 |- dom ( B X. A) (_ B
68	66, 67	sstri 2071	. . . . . 6 |- dom ( f i^i (B X. A)) (_ B
69		ssdom2g 4403	. . . . . 6 |- (B e. V -> (dom ( f i^i (B X. A)) (_ B -> dom ( f i^i (B X. A)) ~<_ B))
70	1, 68, 69	mp2 43	. . . . 5 |- dom ( f i^i (B X. A)) ~<_ B
71		domtr 4409	. . . . 5 |- ((A ~<_ dom ( f i^i (B X. A)) /\ dom ( f i^i (B X. A)) ~<_ B) -> A ~<_ B)
72	70, 71	mpan2 695	. . . 4 |- (A ~<_ dom ( f i^i (B X. A)) -> A ~<_ B)
73	59, 63, 72	3syl 20	. . 3 |- ((A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
74	73	19.23aiv 1295	. 2 |- (E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx) -> A ~<_ B)
75	42, 74	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 953   = wceq 955   e. wcel 957  E.wex 979  E*wmo 1381   =/= wne 1584  A.wral 1644  E.wrex 1645  Vcvv 1809   i^i cin 2044   (_ wss 2045  (/)c0 2278  <.cop 2409   class class class wbr 2616   X. cxp 3165  dom cdm 3167  ran crn 3168  Rel wrel 3172  Fun wfun 3173   Fn wfn 3174  -->wf 3175  -onto->wfo 3177   ~<_ cdom 4362   ~< csdm 4363

This theorem is referenced by:  brdom5 4789  brdom4 4790

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 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-ac 4731

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3