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Theorem ac7g 8845
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 3511 . . . 4  |-  ( x  =  R  ->  (
f  C_  x  <->  f  C_  R ) )
2 dmeq 5192 . . . . 5  |-  ( x  =  R  ->  dom  x  =  dom  R )
32fneq2d 5654 . . . 4  |-  ( x  =  R  ->  (
f  Fn  dom  x  <->  f  Fn  dom  R ) )
41, 3anbi12d 708 . . 3  |-  ( x  =  R  ->  (
( f  C_  x  /\  f  Fn  dom  x )  <->  ( f  C_  R  /\  f  Fn 
dom  R ) ) )
54exbidv 1719 . 2  |-  ( x  =  R  ->  ( E. f ( f  C_  x  /\  f  Fn  dom  x )  <->  E. f
( f  C_  R  /\  f  Fn  dom  R ) ) )
6 ac7 8844 . 2  |-  E. f
( f  C_  x  /\  f  Fn  dom  x )
75, 6vtoclg 3164 1  |-  ( R  e.  A  ->  E. f
( f  C_  R  /\  f  Fn  dom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    C_ wss 3461   dom cdm 4988    Fn wfn 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ac 8488
This theorem is referenced by: (None)
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