MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axcc2 Structured version   Unicode version

Theorem axcc2 8808
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 2624 . . 3  |-  F/_ n if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )
2 nfcv 2624 . . 3  |-  F/_ m if ( ( F `  n )  =  (/) ,  { (/) } ,  ( F `  n ) )
3 fveq2 5859 . . . . 5  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
43eqeq1d 2464 . . . 4  |-  ( m  =  n  ->  (
( F `  m
)  =  (/)  <->  ( F `  n )  =  (/) ) )
54, 3ifbieq2d 3959 . . 3  |-  ( m  =  n  ->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )  =  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
61, 2, 5cbvmpt 4532 . 2  |-  ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) )  =  ( n  e.  om  |->  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
7 nfcv 2624 . . 3  |-  F/_ n
( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )
8 nfcv 2624 . . . 4  |-  F/_ m { n }
9 nffvmpt1 5867 . . . 4  |-  F/_ m
( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n )
108, 9nfxp 5020 . . 3  |-  F/_ m
( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) )
11 sneq 4032 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
12 fveq2 5859 . . . 4  |-  ( m  =  n  ->  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
)  =  ( ( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  n
) )
1311, 12xpeq12d 5019 . . 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 4532 . 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 2624 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) )
16 nfcv 2624 . . . 4  |-  F/_ m 2nd
17 nfcv 2624 . . . . 5  |-  F/_ m
f
18 nffvmpt1 5867 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  n
)
1917, 18nffv 5866 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) )
2016, 19nffv 5866 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) )
21 fveq2 5859 . . . . 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 5863 . . . 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 5863 . . 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 4532 . 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 8807 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 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   A.wral 2809   (/)c0 3780   ifcif 3934   {csn 4022    |-> cmpt 4500    X. cxp 4992    Fn wfn 5576   ` cfv 5581   omcom 6673   2ndc2nd 6775
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cc 8806
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-2nd 6777  df-er 7303  df-en 7509
This theorem is referenced by:  axcc3  8809  acncc  8811  domtriomlem  8813
  Copyright terms: Public domain W3C validator