Theorem ac5b 4725

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 4724	. 2 |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))
3		19.42v 1303	. . 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 3787	. . . . . . . . . 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 3184	. . . . . . . 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 1742	. . . . . . 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 1698	. . . . . . 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 485	. . . 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 1036	. . 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 694	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 955  E.wex 977   =/= wne 1577  A.wral 1637  Vcvv 1802   (_ wss 2037  (/)c0 2270  U.cuni 2493  ran crn 3161   Fn wfn 3167  -->wf 3168  ` cfv 3172

This theorem is referenced by:  ac6lem 4726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188

Copyright terms: Public domain