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Theorem axcc2 8849
Description: A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
Assertion
Ref Expression
axcc2  |-  E. g
( g  Fn  om  /\ 
A. n  e.  om  ( ( F `  n )  =/=  (/)  ->  (
g `  n )  e.  ( F `  n
) ) )
Distinct variable group:    g, F, n

Proof of Theorem axcc2
Dummy variables  f  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2564 . . 3  |-  F/_ n if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )
2 nfcv 2564 . . 3  |-  F/_ m if ( ( F `  n )  =  (/) ,  { (/) } ,  ( F `  n ) )
3 fveq2 5849 . . . . 5  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
43eqeq1d 2404 . . . 4  |-  ( m  =  n  ->  (
( F `  m
)  =  (/)  <->  ( F `  n )  =  (/) ) )
54, 3ifbieq2d 3910 . . 3  |-  ( m  =  n  ->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )  =  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
61, 2, 5cbvmpt 4486 . 2  |-  ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) )  =  ( n  e.  om  |->  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
7 nfcv 2564 . . 3  |-  F/_ n
( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )
8 nfcv 2564 . . . 4  |-  F/_ m { n }
9 nffvmpt1 5857 . . . 4  |-  F/_ m
( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n )
108, 9nfxp 4850 . . 3  |-  F/_ m
( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) )
11 sneq 3982 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
12 fveq2 5849 . . . 4  |-  ( m  =  n  ->  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
)  =  ( ( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  n
) )
1311, 12xpeq12d 4848 . . 3  |-  ( m  =  n  ->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )  =  ( { n }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  n ) ) )
147, 10, 13cbvmpt 4486 . 2  |-  ( m  e.  om  |->  ( { m }  X.  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
) ) )  =  ( n  e.  om  |->  ( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) ) )
15 nfcv 2564 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) )
16 nfcv 2564 . . . 4  |-  F/_ m 2nd
17 nfcv 2564 . . . . 5  |-  F/_ m
f
18 nffvmpt1 5857 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  n
)
1917, 18nffv 5856 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) )
2016, 19nffv 5856 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) )
21 fveq2 5849 . . . . 5  |-  ( m  =  n  ->  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m )  =  ( ( m  e.  om  |->  ( { m }  X.  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
) ) ) `  n ) )
2221fveq2d 5853 . . . 4  |-  ( m  =  n  ->  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) )  =  ( f `
 ( ( m  e.  om  |->  ( { m }  X.  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
) ) ) `  n ) ) )
2322fveq2d 5853 . . 3  |-  ( m  =  n  ->  ( 2nd `  ( f `  ( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  m
) ) )  =  ( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) ) )
2415, 20, 23cbvmpt 4486 . 2  |-  ( m  e.  om  |->  ( 2nd `  ( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) ) )  =  ( n  e.  om  |->  ( 2nd `  ( f `
 ( ( m  e.  om  |->  ( { m }  X.  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
) ) ) `  n ) ) ) )
256, 14, 24axcc2lem 8848 1  |-  E. g
( g  Fn  om  /\ 
A. n  e.  om  ( ( F `  n )  =/=  (/)  ->  (
g `  n )  e.  ( F `  n
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   A.wral 2754   (/)c0 3738   ifcif 3885   {csn 3972    |-> cmpt 4453    X. cxp 4821    Fn wfn 5564   ` cfv 5569   omcom 6683   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cc 8847
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-2nd 6785  df-er 7348  df-en 7555
This theorem is referenced by:  axcc3  8850  acncc  8852  domtriomlem  8854
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