Theorem exfo 3817

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 3815	. . . 4 |- (f:A-onto->B <-> (f:A-->B /\ A.x e. B E.y e. A yfx))
2		dff3 3813	. . . . . 6 |- (f:A-->B <-> (f (_ (A X. B) /\ A.x e. A E!y e. B xfy))
3	2	pm3.27bi 326	. . . . 5 |- (f:A-->B -> A.x e. A E!y e. B xfy)
4	3	anim1i 334	. . . 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 199	. . 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	19.22i 1039	. 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 3228	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B) -> (xfy <-> x(f i^i (A X. B))y))
8	7	reubidva 1777	. . . . . . . . . . 11 |- (x e. A -> (E!y e. B xfy <-> E!y e. B x(f i^i (A X. B))y))
9	8	biimpd 153	. . . . . . . . . 10 |- (x e. A -> (E!y e. B xfy -> E!y e. B x(f i^i (A X. B))y))
10	9	r19.20i 1702	. . . . . . . . 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 2228	. . . . . . . . 9 |- (f i^i (A X. B)) (_ (A X. B)
12	10, 11	jctil 292	. . . . . . . 8 |- (A.x e. A E!y e. B xfy -> ((f i^i (A X. B)) (_ (A X. B) /\ A.x e. A E!y e. B x(f i^i (A X. B))y))
13		dff3 3813	. . . . . . . 8 |- ((f i^i (A X. B)):A-->B <-> ((f i^i (A X. B)) (_ (A X. B) /\ A.x e. A E!y e. B x(f i^i (A X. B))y))
14	12, 13	sylibr 200	. . . . . . 7 |- (A.x e. A E!y e. B xfy -> (f i^i (A X. B)):A-->B)
15		rninxp 3478	. . . . . . . 8 |- (ran ( f i^i (A X. B)) = B <-> A.x e. B E.y e. A yfx)
16	15	biimpr 152	. . . . . . 7 |- (A.x e. B E.y e. A yfx -> ran ( f i^i (A X. B)) = B)
17	14, 16	anim12i 333	. . . . . 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 3670	. . . . . 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 200	. . . . 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 1810	. . . . . . 7 |- f e. V
21	20	inex1 2712	. . . . . 6 |- (f i^i (A X. B)) e. V
22		foeq1 3663	. . . . . 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 1866	. . . . 5 |- ((f i^i (A X. B)):A-onto->B -> E.g g:A-onto->B)
24	19, 23	syl 10	. . . 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 1294	. . 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 3663	. . . 4 |- (g = f -> (g:A-onto->B <-> f:A-onto->B))
27	26	cbvexv 1314	. . 3 |- (E.g g:A-onto->B <-> E.f f:A-onto->B)
28	25, 27	sylib 198	. 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	impbi 157	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 146   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  A.wral 1643  E.wrex 1644  E!wreu 1645   i^i cin 2043   (_ wss 2044   class class class wbr 2615   X. cxp 3164  ran crn 3167  -->wf 3174  -onto->wfo 3176

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 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fo 3192  df-fv 3194

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