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Theorem axdc 8796
Description: This theorem derives ax-dc 8721 using ax-ac 8734 and ax-inf 7950. 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 4399 . . . . . . . . 9  |-  ( w  =  z  ->  (
u x w  <->  u x
z ) )
21cbvabv 2595 . . . . . . . 8  |-  { w  |  u x w }  =  { z  |  u x z }
3 breq1 4398 . . . . . . . . 9  |-  ( u  =  v  ->  (
u x z  <->  v x
z ) )
43abbidv 2588 . . . . . . . 8  |-  ( u  =  v  ->  { z  |  u x z }  =  { z  |  v x z } )
52, 4syl5eq 2505 . . . . . . 7  |-  ( u  =  v  ->  { w  |  u x w }  =  { z  |  v x z } )
65fveq2d 5798 . . . . . 6  |-  ( u  =  v  ->  (
g `  { w  |  u x w }
)  =  ( g `
 { z  |  v x z } ) )
76cbvmptv 4486 . . . . 5  |-  ( u  e.  _V  |->  ( g `
 { w  |  u x w }
) )  =  ( v  e.  _V  |->  ( g `  { z  |  v x z } ) )
8 rdgeq1 6972 . . . . 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 5207 . . . . 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 8795 . . 3  |-  ( E. z  y x z  ->  ( ran  x  C_ 
dom  x  ->  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
) )
1211exlimiv 1689 . 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 1370   E.wex 1587   {cab 2437   A.wral 2796   _Vcvv 3072    C_ wss 3431   class class class wbr 4395    |-> cmpt 4453   suc csuc 4824   dom cdm 4943   ran crn 4944    |` cres 4945   ` cfv 5521   omcom 6581   reccrdg 6970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-ac2 8738
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-om 6582  df-recs 6937  df-rdg 6971  df-ac 8392
This theorem is referenced by: (None)
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