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Theorem ac7g 8749
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 3481 . . . 4  |-  ( x  =  R  ->  (
f  C_  x  <->  f  C_  R ) )
2 dmeq 5143 . . . . 5  |-  ( x  =  R  ->  dom  x  =  dom  R )
32fneq2d 5605 . . . 4  |-  ( x  =  R  ->  (
f  Fn  dom  x  <->  f  Fn  dom  R ) )
41, 3anbi12d 710 . . 3  |-  ( x  =  R  ->  (
( f  C_  x  /\  f  Fn  dom  x )  <->  ( f  C_  R  /\  f  Fn 
dom  R ) ) )
54exbidv 1681 . 2  |-  ( x  =  R  ->  ( E. f ( f  C_  x  /\  f  Fn  dom  x )  <->  E. f
( f  C_  R  /\  f  Fn  dom  R ) ) )
6 ac7 8748 . 2  |-  E. f
( f  C_  x  /\  f  Fn  dom  x )
75, 6vtoclg 3130 1  |-  ( R  e.  A  ->  E. f
( f  C_  R  /\  f  Fn  dom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    C_ wss 3431   dom cdm 4943    Fn wfn 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-ac2 8738
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ac 8392
This theorem is referenced by: (None)
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