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Theorem axdc 8814
Description: This theorem derives ax-dc 8739 using ax-ac 8752 and ax-inf 7969. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
axdc  |-  ( ( E. y E. z 
y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )
Distinct variable group:    f, n, x, y, z

Proof of Theorem axdc
Dummy variables  v 
g  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4371 . . . . . . . . 9  |-  ( w  =  z  ->  (
u x w  <->  u x
z ) )
21cbvabv 2525 . . . . . . . 8  |-  { w  |  u x w }  =  { z  |  u x z }
3 breq1 4370 . . . . . . . . 9  |-  ( u  =  v  ->  (
u x z  <->  v x
z ) )
43abbidv 2518 . . . . . . . 8  |-  ( u  =  v  ->  { z  |  u x z }  =  { z  |  v x z } )
52, 4syl5eq 2435 . . . . . . 7  |-  ( u  =  v  ->  { w  |  u x w }  =  { z  |  v x z } )
65fveq2d 5778 . . . . . 6  |-  ( u  =  v  ->  (
g `  { w  |  u x w }
)  =  ( g `
 { z  |  v x z } ) )
76cbvmptv 4458 . . . . 5  |-  ( u  e.  _V  |->  ( g `
 { w  |  u x w }
) )  =  ( v  e.  _V  |->  ( g `  { z  |  v x z } ) )
8 rdgeq1 6995 . . . . 5  |-  ( ( u  e.  _V  |->  ( g `  { w  |  u x w }
) )  =  ( v  e.  _V  |->  ( g `  { z  |  v x z } ) )  ->  rec ( ( u  e. 
_V  |->  ( g `  { w  |  u x w } ) ) ,  y )  =  rec ( ( v  e.  _V  |->  ( g `  { z  |  v x z } ) ) ,  y ) )
9 reseq1 5180 . . . . 5  |-  ( rec ( ( u  e. 
_V  |->  ( g `  { w  |  u x w } ) ) ,  y )  =  rec ( ( v  e.  _V  |->  ( g `  { z  |  v x z } ) ) ,  y )  ->  ( rec ( ( u  e. 
_V  |->  ( g `  { w  |  u x w } ) ) ,  y )  |`  om )  =  ( rec ( ( v  e.  _V  |->  ( g `
 { z  |  v x z } ) ) ,  y )  |`  om )
)
107, 8, 9mp2b 10 . . . 4  |-  ( rec ( ( u  e. 
_V  |->  ( g `  { w  |  u x w } ) ) ,  y )  |`  om )  =  ( rec ( ( v  e.  _V  |->  ( g `
 { z  |  v x z } ) ) ,  y )  |`  om )
1110axdclem2 8813 . . 3  |-  ( E. z  y x z  ->  ( ran  x  C_ 
dom  x  ->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
) )
1211exlimiv 1730 . 2  |-  ( E. y E. z  y x z  ->  ( ran  x  C_  dom  x  ->  E. f A. n  e. 
om  ( f `  n ) x ( f `  suc  n
) ) )
1312imp 427 1  |-  ( ( E. y E. z 
y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   E.wex 1620   {cab 2367   A.wral 2732   _Vcvv 3034    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   suc csuc 4794   dom cdm 4913   ran crn 4914    |` cres 4915   ` cfv 5496   omcom 6599   reccrdg 6993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-ac2 8756
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994  df-ac 8410
This theorem is referenced by: (None)
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