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Theorem axcc2 8063
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 2419 . . 3  |-  F/_ n if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )
2 nfcv 2419 . . 3  |-  F/_ m if ( ( F `  n )  =  (/) ,  { (/) } ,  ( F `  n ) )
3 fveq2 5525 . . . . 5  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
43eqeq1d 2291 . . . 4  |-  ( m  =  n  ->  (
( F `  m
)  =  (/)  <->  ( F `  n )  =  (/) ) )
54, 3ifbieq2d 3585 . . 3  |-  ( m  =  n  ->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )  =  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
61, 2, 5cbvmpt 4110 . 2  |-  ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) )  =  ( n  e.  om  |->  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
7 nfcv 2419 . . 3  |-  F/_ n
( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )
8 nfcv 2419 . . . 4  |-  F/_ m { n }
9 nfmpt1 4109 . . . . 5  |-  F/_ m
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) )
10 nfcv 2419 . . . . 5  |-  F/_ m n
119, 10nffv 5532 . . . 4  |-  F/_ m
( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n )
128, 11nfxp 4715 . . 3  |-  F/_ m
( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) )
13 sneq 3651 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
14 fveq2 5525 . . . 4  |-  ( m  =  n  ->  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
)  =  ( ( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  n
) )
1513, 14xpeq12d 4714 . . 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 ) ) )
167, 12, 15cbvmpt 4110 . 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 ) ) )
17 nfcv 2419 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) )
18 nfcv 2419 . . . 4  |-  F/_ m 2nd
19 nfcv 2419 . . . . 5  |-  F/_ m
f
20 nfmpt1 4109 . . . . . 6  |-  F/_ m
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) )
2120, 10nffv 5532 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  n
)
2219, 21nffv 5532 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) )
2318, 22nffv 5532 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) )
24 fveq2 5525 . . . . 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 ) )
2524fveq2d 5529 . . . 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 ) ) )
2625fveq2d 5529 . . 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 ) ) ) )
2717, 23, 26cbvmpt 4110 . 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 ) ) ) )
286, 16, 27axcc2lem 8062 1  |-  E. g
( g  Fn  om  /\ 
A. n  e.  om  ( ( F `  n )  =/=  (/)  ->  (
g `  n )  e.  ( F `  n
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   (/)c0 3455   ifcif 3565   {csn 3640    e. cmpt 4077   omcom 4656    X. cxp 4687    Fn wfn 5250   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  axcc3  8064  acncc  8066  domtriomlem  8068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-2nd 6123  df-er 6660  df-en 6864
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