Theorem exfo 3822

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 3820	. . . 4 |- (f:A-onto->B <-> (f:A-->B /\ A.x e. B E.y e. A yfx))
2		dff3 3818	. . . . . 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 1040	. 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 3232	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B) -> (xfy <-> x(f i^i (A X. B))y))
8	7	reubidva 1779	. . . . . . . . . . 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 1704	. . . . . . . . 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 2231	. . . . . . . . 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 3818	. . . . . . . 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 3482	. . . . . . . 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 3675	. . . . . 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 1813	. . . . . . 7 |- f e. V
21	20	inex1 2716	. . . . . 6 |- (f i^i (A X. B)) e. V
22		foeq1 3668	. . . . . 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 1869	. . . . 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 1295	. . 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 3668	. . . 4 |- (g = f -> (g:A-onto->B <-> f:A-onto->B))
27	26	cbvexv 1315	. . 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 956   e. wcel 958  E.wex 980  A.wral 1645  E.wrex 1646  E!wreu 1647   i^i cin 2046   (_ wss 2047   class class class wbr 2619   X. cxp 3168  ran crn 3171  -->wf 3178  -onto->wfo 3180

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198

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