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Theorem ac5 7988
 Description: An Axiom of Choice equivalent: there exists a function (called a choice function) with domain that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that be a function is not necessary; see ac4 7986. (Contributed by NM, 29-Aug-1999.)
Hypothesis
Ref Expression
ac5.1
Assertion
Ref Expression
ac5
Distinct variable group:   ,,

Proof of Theorem ac5
StepHypRef Expression
1 ac5.1 . 2
2 fneq2 5191 . . . 4
3 raleq 2689 . . . 4
42, 3anbi12d 694 . . 3
54exbidv 2005 . 2
6 dfac4 7633 . . 3 CHOICE
76axaci 7979 . 2
81, 5, 7vtocl 2776 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360  wex 1537   wceq 1619   wcel 1621   wne 2412  wral 2509  cvv 2727  c0 3362   wfn 4587  cfv 4592 This theorem is referenced by:  ac5g  24240 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ac 7627
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