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Theorem axdc 8686
Description: This theorem derives ax-dc 8611 using ax-ac 8624 and ax-inf 7840. 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 4293 . . . . . . . . 9  |-  ( w  =  z  ->  (
u x w  <->  u x
z ) )
21cbvabv 2560 . . . . . . . 8  |-  { w  |  u x w }  =  { z  |  u x z }
3 breq1 4292 . . . . . . . . 9  |-  ( u  =  v  ->  (
u x z  <->  v x
z ) )
43abbidv 2555 . . . . . . . 8  |-  ( u  =  v  ->  { z  |  u x z }  =  { z  |  v x z } )
52, 4syl5eq 2485 . . . . . . 7  |-  ( u  =  v  ->  { w  |  u x w }  =  { z  |  v x z } )
65fveq2d 5692 . . . . . 6  |-  ( u  =  v  ->  (
g `  { w  |  u x w }
)  =  ( g `
 { z  |  v x z } ) )
76cbvmptv 4380 . . . . 5  |-  ( u  e.  _V  |->  ( g `
 { w  |  u x w }
) )  =  ( v  e.  _V  |->  ( g `  { z  |  v x z } ) )
8 rdgeq1 6863 . . . . 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 5100 . . . . 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 8685 . . 3  |-  ( E. z  y x z  ->  ( ran  x  C_ 
dom  x  ->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
) )
1211exlimiv 1693 . 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 429 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 369    = wceq 1364   E.wex 1591   {cab 2427   A.wral 2713   _Vcvv 2970    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   suc csuc 4717   dom cdm 4836   ran crn 4837    |` cres 4838   ` cfv 5415   omcom 6475   reccrdg 6861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-ac 8282
This theorem is referenced by: (None)
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