MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac7g Unicode version

Theorem ac7g 8343
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 3362 . . . 4  |-  ( x  =  R  ->  (
f  C_  x  <->  f  C_  R ) )
2 dmeq 5061 . . . . 5  |-  ( x  =  R  ->  dom  x  =  dom  R )
32fneq2d 5528 . . . 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 1636 . 2  |-  ( x  =  R  ->  ( E. f ( f  C_  x  /\  f  Fn  dom  x )  <->  E. f
( f  C_  R  /\  f  Fn  dom  R ) ) )
6 ac7 8342 . 2  |-  E. f
( f  C_  x  /\  f  Fn  dom  x )
75, 6vtoclg 3003 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 1550    = wceq 1652    e. wcel 1725    C_ wss 3312   dom cdm 4869    Fn wfn 5440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-ac2 8332
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ac 7986
  Copyright terms: Public domain W3C validator