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Theorem ac7g 8310
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 3330 . . . 4  |-  ( x  =  R  ->  (
f  C_  x  <->  f  C_  R ) )
2 dmeq 5029 . . . . 5  |-  ( x  =  R  ->  dom  x  =  dom  R )
32fneq2d 5496 . . . 4  |-  ( x  =  R  ->  (
f  Fn  dom  x  <->  f  Fn  dom  R ) )
41, 3anbi12d 692 . . 3  |-  ( x  =  R  ->  (
( f  C_  x  /\  f  Fn  dom  x )  <->  ( f  C_  R  /\  f  Fn 
dom  R ) ) )
54exbidv 1633 . 2  |-  ( x  =  R  ->  ( E. f ( f  C_  x  /\  f  Fn  dom  x )  <->  E. f
( f  C_  R  /\  f  Fn  dom  R ) ) )
6 ac7 8309 . 2  |-  E. f
( f  C_  x  /\  f  Fn  dom  x )
75, 6vtoclg 2971 1  |-  ( R  e.  A  ->  E. f
( f  C_  R  /\  f  Fn  dom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    C_ wss 3280   dom cdm 4837    Fn wfn 5408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-ac2 8299
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ac 7953
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