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

Theorem axcc2 8317
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 2572 . . 3  |-  F/_ n if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )
2 nfcv 2572 . . 3  |-  F/_ m if ( ( F `  n )  =  (/) ,  { (/) } ,  ( F `  n ) )
3 fveq2 5728 . . . . 5  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
43eqeq1d 2444 . . . 4  |-  ( m  =  n  ->  (
( F `  m
)  =  (/)  <->  ( F `  n )  =  (/) ) )
54, 3ifbieq2d 3759 . . 3  |-  ( m  =  n  ->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )  =  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
61, 2, 5cbvmpt 4299 . 2  |-  ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) )  =  ( n  e.  om  |->  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
7 nfcv 2572 . . 3  |-  F/_ n
( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )
8 nfcv 2572 . . . 4  |-  F/_ m { n }
9 nffvmpt1 5736 . . . 4  |-  F/_ m
( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n )
108, 9nfxp 4904 . . 3  |-  F/_ m
( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) )
11 sneq 3825 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
12 fveq2 5728 . . . 4  |-  ( m  =  n  ->  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
)  =  ( ( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  n
) )
1311, 12xpeq12d 4903 . . 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 4299 . 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 2572 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) )
16 nfcv 2572 . . . 4  |-  F/_ m 2nd
17 nfcv 2572 . . . . 5  |-  F/_ m
f
18 nffvmpt1 5736 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  n
)
1917, 18nffv 5735 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) )
2016, 19nffv 5735 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) )
21 fveq2 5728 . . . . 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 5732 . . . 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 5732 . . 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 4299 . 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 8316 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 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   (/)c0 3628   ifcif 3739   {csn 3814    e. cmpt 4266   omcom 4845    X. cxp 4876    Fn wfn 5449   ` cfv 5454   2ndc2nd 6348
This theorem is referenced by:  axcc3  8318  acncc  8320  domtriomlem  8322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cc 8315
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-2nd 6350  df-er 6905  df-en 7110
  Copyright terms: Public domain W3C validator