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Theorem cc2 7378
Description: Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
Hypotheses
Ref Expression
cc2.cc  |-  ( ph  -> CCHOICE )
cc2.a  |-  ( ph  ->  F  Fn  om )
cc2.m  |-  ( ph  ->  A. x  e.  om  E. w  w  e.  ( F `  x ) )
Assertion
Ref Expression
cc2  |-  ( ph  ->  E. g ( g  Fn  om  /\  A. n  e.  om  (
g `  n )  e.  ( F `  n
) ) )
Distinct variable groups:    g, F, n   
w, F, x    ph, n
Allowed substitution hints:    ph( x, w, g)

Proof of Theorem cc2
Dummy variables  f  m  v  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cc2.cc . 2  |-  ( ph  -> CCHOICE )
2 cc2.a . 2  |-  ( ph  ->  F  Fn  om )
3 cc2.m . . . 4  |-  ( ph  ->  A. x  e.  om  E. w  w  e.  ( F `  x ) )
4 fveq2 5575 . . . . . . 7  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
54eleq2d 2274 . . . . . 6  |-  ( x  =  y  ->  (
w  e.  ( F `
 x )  <->  w  e.  ( F `  y ) ) )
65exbidv 1847 . . . . 5  |-  ( x  =  y  ->  ( E. w  w  e.  ( F `  x )  <->  E. w  w  e.  ( F `  y ) ) )
76cbvralv 2737 . . . 4  |-  ( A. x  e.  om  E. w  w  e.  ( F `  x )  <->  A. y  e.  om  E. w  w  e.  ( F `  y ) )
83, 7sylib 122 . . 3  |-  ( ph  ->  A. y  e.  om  E. w  w  e.  ( F `  y ) )
9 eleq1w 2265 . . . . 5  |-  ( w  =  v  ->  (
w  e.  ( F `
 y )  <->  v  e.  ( F `  y ) ) )
109cbvexv 1941 . . . 4  |-  ( E. w  w  e.  ( F `  y )  <->  E. v  v  e.  ( F `  y ) )
1110ralbii 2511 . . 3  |-  ( A. y  e.  om  E. w  w  e.  ( F `  y )  <->  A. y  e.  om  E. v  v  e.  ( F `  y ) )
128, 11sylib 122 . 2  |-  ( ph  ->  A. y  e.  om  E. v  v  e.  ( F `  y ) )
13 nfcv 2347 . . 3  |-  F/_ n
( { m }  X.  ( F `  m
) )
14 nfcv 2347 . . 3  |-  F/_ m
( { n }  X.  ( F `  n
) )
15 sneq 3643 . . . 4  |-  ( m  =  n  ->  { m }  =  { n } )
16 fveq2 5575 . . . 4  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
1715, 16xpeq12d 4699 . . 3  |-  ( m  =  n  ->  ( { m }  X.  ( F `  m ) )  =  ( { n }  X.  ( F `  n )
) )
1813, 14, 17cbvmpt 4138 . 2  |-  ( m  e.  om  |->  ( { m }  X.  ( F `  m )
) )  =  ( n  e.  om  |->  ( { n }  X.  ( F `  n ) ) )
19 nfcv 2347 . . 3  |-  F/_ n
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  m
) ) )
20 nfcv 2347 . . . 4  |-  F/_ m 2nd
21 nfcv 2347 . . . . 5  |-  F/_ m
f
22 nffvmpt1 5586 . . . . 5  |-  F/_ m
( ( m  e. 
om  |->  ( { m }  X.  ( F `  m ) ) ) `
 n )
2321, 22nffv 5585 . . . 4  |-  F/_ m
( f `  (
( m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  n
) )
2420, 23nffv 5585 . . 3  |-  F/_ m
( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  n
) ) )
25 2fveq3 5580 . . . 4  |-  ( m  =  n  ->  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  m
) )  =  ( f `  ( ( m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  n
) ) )
2625fveq2d 5579 . . 3  |-  ( m  =  n  ->  ( 2nd `  ( f `  ( ( m  e. 
om  |->  ( { m }  X.  ( F `  m ) ) ) `
 m ) ) )  =  ( 2nd `  ( f `  (
( m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  n
) ) ) )
2719, 24, 26cbvmpt 4138 . 2  |-  ( m  e.  om  |->  ( 2nd `  ( f `  (
( m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  m
) ) ) )  =  ( n  e. 
om  |->  ( 2nd `  (
f `  ( (
m  e.  om  |->  ( { m }  X.  ( F `  m ) ) ) `  n
) ) ) )
281, 2, 12, 18, 27cc2lem 7377 1  |-  ( ph  ->  E. g ( g  Fn  om  /\  A. n  e.  om  (
g `  n )  e.  ( F `  n
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1514    e. wcel 2175   A.wral 2483   {csn 3632    |-> cmpt 4104   omcom 4637    X. cxp 4672    Fn wfn 5265   ` cfv 5270   2ndc2nd 6224  CCHOICEwacc 7373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-iinf 4635
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-2nd 6226  df-er 6619  df-en 6827  df-cc 7374
This theorem is referenced by:  cc3  7379  acnccim  7383
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