Theorem exfo 4795

Description: A relation equivalent to the existence of an onto mapping. The right-hand f is not necessarily a function.

Assertion

Ref	Expression
exfo	|- (E.f f:A-onto->B <-> E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))

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

Proof of Theorem exfo

Step	Hyp	Ref	Expression
1		dffo4 4793	. . . 4 |- (f:A-onto->B <-> (f:A-->B /\ A.x e. B E.y e. A yfx))
2		dff4 4791	. . . . . 6 |- (f:A-->B <-> (f C_ (A X. B) /\ A.x e. A E!y e. B xfy))
3	2	simprbi 353	. . . . 5 |- (f:A-->B -> A.x e. A E!y e. B xfy)
4	3	anim1i 361	. . . 4 |- ((f:A-->B /\ A.x e. B E.y e. A yfx) -> (A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))
5	1, 4	sylbi 216	. . 3 |- (f:A-onto->B -> (A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))
6	5	eximi 1387	. 2 |- (E.f f:A-onto->B -> E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))
7		brinxp 4058	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B) -> (xfy <-> x(f i^i (A X. B))y))
8	7	reubidva 2259	. . . . . . . . . . 11 |- (x e. A -> (E!y e. B xfy <-> E!y e. B x(f i^i (A X. B))y))
9	8	biimpd 170	. . . . . . . . . 10 |- (x e. A -> (E!y e. B xfy -> E!y e. B x(f i^i (A X. B))y))
10	9	ralimia 2166	. . . . . . . . 9 |- (A.x e. A E!y e. B xfy -> A.x e. A E!y e. B x(f i^i (A X. B))y)
11		inss2 2813	. . . . . . . . 9 |- (f i^i (A X. B)) C_ (A X. B)
12	10, 11	jctil 316	. . . . . . . 8 |- (A.x e. A E!y e. B xfy -> ((f i^i (A X. B)) C_ (A X. B) /\ A.x e. A E!y e. B x(f i^i (A X. B))y))
13		dff4 4791	. . . . . . . 8 |- ((f i^i (A X. B)):A-->B <-> ((f i^i (A X. B)) C_ (A X. B) /\ A.x e. A E!y e. B x(f i^i (A X. B))y))
14	12, 13	sylibr 217	. . . . . . 7 |- (A.x e. A E!y e. B xfy -> (f i^i (A X. B)):A-->B)
15		rninxp 4355	. . . . . . . 8 |- (ran ( f i^i (A X. B)) = B <-> A.x e. B E.y e. A yfx)
16	15	biimpri 169	. . . . . . 7 |- (A.x e. B E.y e. A yfx -> ran ( f i^i (A X. B)) = B)
17	14, 16	anim12i 360	. . . . . 6 |- ((A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx) -> ((f i^i (A X. B)):A-->B /\ ran ( f i^i (A X. B)) = B))
18		dffo2 4621	. . . . . 6 |- ((f i^i (A X. B)):A-onto->B <-> ((f i^i (A X. B)):A-->B /\ ran ( f i^i (A X. B)) = B))
19	17, 18	sylibr 217	. . . . 5 |- ((A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx) -> (f i^i (A X. B)):A-onto->B)
20		visset 2295	. . . . . . 7 |- f e. _V
21	20	inex1 3452	. . . . . 6 |- (f i^i (A X. B)) e. _V
22		foeq1 4613	. . . . . 6 |- (g = (f i^i (A X. B)) -> (g:A-onto->B <-> (f i^i (A X. B)):A-onto->B))
23	21, 22	cla4ev 2371	. . . . 5 |- ((f i^i (A X. B)):A-onto->B -> E.g g:A-onto->B)
24	19, 23	syl 12	. . . 4 |- ((A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx) -> E.g g:A-onto->B)
25	24	19.23aiv 1674	. . 3 |- (E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx) -> E.g g:A-onto->B)
26		foeq1 4613	. . . 4 |- (g = f -> (g:A-onto->B <-> f:A-onto->B))
27	26	cbvexv 1697	. . 3 |- (E.g g:A-onto->B <-> E.f f:A-onto->B)
28	25, 27	sylib 215	. 2 |- (E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx) -> E.f f:A-onto->B)
29	6, 28	impbii 174	1 |- (E.f f:A-onto->B <-> E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  E!wreu 2107   i^i cin 2592   C_ wss 2593   class class class wbr 3338   X. cxp 3984  ran crn 3987  -->wf 3994  -onto->wfo 3996

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

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-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-fo 4012  df-fv 4014

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