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Theorem axcc2 8079
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 2432 . . 3  |-  F/_ n if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )
2 nfcv 2432 . . 3  |-  F/_ m if ( ( F `  n )  =  (/) ,  { (/) } ,  ( F `  n ) )
3 fveq2 5541 . . . . 5  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
43eqeq1d 2304 . . . 4  |-  ( m  =  n  ->  (
( F `  m
)  =  (/)  <->  ( F `  n )  =  (/) ) )
54, 3ifbieq2d 3598 . . 3  |-  ( m  =  n  ->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) )  =  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
61, 2, 5cbvmpt 4126 . 2  |-  ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) )  =  ( n  e.  om  |->  if ( ( F `  n
)  =  (/) ,  { (/)
} ,  ( F `
 n ) ) )
7 nfcv 2432 . . 3  |-  F/_ n
( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) )
8 nfcv 2432 . . . 4  |-  F/_ m { n }
9 nfmpt1 4125 . . . . 5  |-  F/_ m
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) )
10 nfcv 2432 . . . . 5  |-  F/_ m n
119, 10nffv 5548 . . . 4  |-  F/_ m
( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n )
128, 11nfxp 4731 . . 3  |-  F/_ m
( { n }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 n ) )
13 sneq 3664 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
14 fveq2 5541 . . . 4  |-  ( m  =  n  ->  (
( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  m
)  =  ( ( m  e.  om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `  n
) )
1513, 14xpeq12d 4730 . . 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 4126 . 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 2432 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  m ) ) )
18 nfcv 2432 . . . 4  |-  F/_ m 2nd
19 nfcv 2432 . . . . 5  |-  F/_ m
f
20 nfmpt1 4125 . . . . . 6  |-  F/_ m
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) )
2120, 10nffv 5548 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( ( m  e.  om  |->  if ( ( F `  m
)  =  (/) ,  { (/)
} ,  ( F `
 m ) ) ) `  m ) ) ) `  n
)
2219, 21nffv 5548 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) )
2318, 22nffv 5548 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( ( m  e. 
om  |->  if ( ( F `  m )  =  (/) ,  { (/) } ,  ( F `  m ) ) ) `
 m ) ) ) `  n ) ) )
24 fveq2 5541 . . . . 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 5545 . . . 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 5545 . . 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 4126 . 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 8078 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ifcif 3578   {csn 3653    e. cmpt 4093   omcom 4672    X. cxp 4703    Fn wfn 5266   ` cfv 5271   2ndc2nd 6137
This theorem is referenced by:  axcc3  8080  acncc  8082  domtriomlem  8084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-2nd 6139  df-er 6676  df-en 6880
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