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Theorem ac7g 8101
Description: An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
ac7g  |-  ( R  e.  A  ->  E. f
( f  C_  R  /\  f  Fn  dom  R ) )
Distinct variable group:    R, f
Allowed substitution hint:    A( f)

Proof of Theorem ac7g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3200 . . . 4  |-  ( x  =  R  ->  (
f  C_  x  <->  f  C_  R ) )
2 dmeq 4879 . . . . 5  |-  ( x  =  R  ->  dom  x  =  dom  R )
32fneq2d 5336 . . . 4  |-  ( x  =  R  ->  (
f  Fn  dom  x  <->  f  Fn  dom  R ) )
41, 3anbi12d 691 . . 3  |-  ( x  =  R  ->  (
( f  C_  x  /\  f  Fn  dom  x )  <->  ( f  C_  R  /\  f  Fn 
dom  R ) ) )
54exbidv 1612 . 2  |-  ( x  =  R  ->  ( E. f ( f  C_  x  /\  f  Fn  dom  x )  <->  E. f
( f  C_  R  /\  f  Fn  dom  R ) ) )
6 ac7 8100 . 2  |-  E. f
( f  C_  x  /\  f  Fn  dom  x )
75, 6vtoclg 2843 1  |-  ( R  e.  A  ->  E. f
( f  C_  R  /\  f  Fn  dom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689    Fn wfn 5250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ac 7743
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