Theorem ac5b 4753

Description: Equivalent of Axiom of Choice.

Hypothesis

Ref	Expression
ac5b.1	|- A e. V

Assertion

Ref	Expression
ac5b	|- (A.x e. A x =/= (/) -> E.f(f:A-->U.A /\ A.x e. A (f` x) e. x))

Distinct variable group:   x,f,A

Proof of Theorem ac5b

Step	Hyp	Ref	Expression
1		ac5b.1	. . 3 |- A e. V
2	1	ac5 4752	. 2 |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))
3		19.42v 1308	. . 3 |- (E.f(A.x e. A x =/= (/) /\ (f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))) <-> (A.x e. A x =/= (/) /\ E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))))
4		chfnrn 3802	. . . . . . . . . 10 |- ((f Fn A /\ A.x e. A (f` x) e. x) -> ran f (_ U.A)
5	4	ex 373	. . . . . . . . 9 |- (f Fn A -> (A.x e. A (f` x) e. x -> ran f (_ U.A))
6	5	anc2li 302	. . . . . . . 8 |- (f Fn A -> (A.x e. A (f` x) e. x -> (f Fn A /\ ran f (_ U.A)))
7		df-f 3194	. . . . . . . 8 |- (f:A-->U.A <-> (f Fn A /\ ran f (_ U.A))
8	6, 7	syl6ibr 213	. . . . . . 7 |- (f Fn A -> (A.x e. A (f` x) e. x -> f:A-->U.A))
9	8	impac 387	. . . . . 6 |- ((f Fn A /\ A.x e. A (f` x) e. x) -> (f:A-->U.A /\ A.x e. A (f` x) e. x))
10		r19.26 1750	. . . . . . 7 |- (A.x e. A (x =/= (/) /\ (x =/= (/) -> (f` x) e. x)) <-> (A.x e. A x =/= (/) /\ A.x e. A (x =/= (/) -> (f` x) e. x)))
11		pm3.35 359	. . . . . . . 8 |- ((x =/= (/) /\ (x =/= (/) -> (f` x) e. x)) -> (f` x) e. x)
12	11	r19.20si 1706	. . . . . . 7 |- (A.x e. A (x =/= (/) /\ (x =/= (/) -> (f` x) e. x)) -> A.x e. A (f` x) e. x)
13	10, 12	sylbir 201	. . . . . 6 |- ((A.x e. A x =/= (/) /\ A.x e. A (x =/= (/) -> (f` x) e. x)) -> A.x e. A (f` x) e. x)
14	9, 13	sylan2 451	. . . . 5 |- ((f Fn A /\ (A.x e. A x =/= (/) /\ A.x e. A (x =/= (/) -> (f` x) e. x))) -> (f:A-->U.A /\ A.x e. A (f` x) e. x))
15	14	an1s 486	. . . 4 |- ((A.x e. A x =/= (/) /\ (f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))) -> (f:A-->U.A /\ A.x e. A (f` x) e. x))
16	15	19.22i 1040	. . 3 |- (E.f(A.x e. A x =/= (/) /\ (f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))) -> E.f(f:A-->U.A /\ A.x e. A (f` x) e. x))
17	3, 16	sylbir 201	. 2 |- ((A.x e. A x =/= (/) /\ E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))) -> E.f(f:A-->U.A /\ A.x e. A (f` x) e. x))
18	2, 17	mpan2 696	1 |- (A.x e. A x =/= (/) -> E.f(f:A-->U.A /\ A.x e. A (f` x) e. x))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  Vcvv 1811   (_ wss 2047  (/)c0 2280  U.cuni 2503  ran crn 3171   Fn wfn 3177  -->wf 3178  ` cfv 3182

This theorem is referenced by:  ac6lem 4754

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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1652  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-fv 3198

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