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Theorem cc4f 7089
Description: Countable choice by showing the existence of a function 
f which can choose a value at each index 
n such that  ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
Hypotheses
Ref Expression
cc4f.cc  |-  ( ph  -> CCHOICE )
cc4f.1  |-  ( ph  ->  A  e.  V )
cc4f.a  |-  F/_ n A
cc4f.2  |-  ( ph  ->  N  ~~  om )
cc4f.3  |-  ( x  =  ( f `  n )  ->  ( ps 
<->  ch ) )
cc4f.m  |-  ( ph  ->  A. n  e.  N  E. x  e.  A  ps )
Assertion
Ref Expression
cc4f  |-  ( ph  ->  E. f ( f : N --> A  /\  A. n  e.  N  ch ) )
Distinct variable groups:    A, f, x   
f, N, n    ch, x    ph, f, n    ps, f    x, n
Allowed substitution hints:    ph( x)    ps( x, n)    ch( f, n)    A( n)    N( x)    V( x, f, n)

Proof of Theorem cc4f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 cc4f.cc . . 3  |-  ( ph  -> CCHOICE )
2 cc4f.1 . . . . 5  |-  ( ph  ->  A  e.  V )
3 rabexg 4071 . . . . 5  |-  ( A  e.  V  ->  { x  e.  A  |  ps }  e.  _V )
42, 3syl 14 . . . 4  |-  ( ph  ->  { x  e.  A  |  ps }  e.  _V )
54ralrimivw 2506 . . 3  |-  ( ph  ->  A. n  e.  N  { x  e.  A  |  ps }  e.  _V )
6 cc4f.m . . . 4  |-  ( ph  ->  A. n  e.  N  E. x  e.  A  ps )
7 rabn0m 3390 . . . . 5  |-  ( E. w  w  e.  {
x  e.  A  |  ps }  <->  E. x  e.  A  ps )
87ralbii 2441 . . . 4  |-  ( A. n  e.  N  E. w  w  e.  { x  e.  A  |  ps } 
<-> 
A. n  e.  N  E. x  e.  A  ps )
96, 8sylibr 133 . . 3  |-  ( ph  ->  A. n  e.  N  E. w  w  e.  { x  e.  A  |  ps } )
10 cc4f.2 . . 3  |-  ( ph  ->  N  ~~  om )
111, 5, 9, 10cc3 7088 . 2  |-  ( ph  ->  E. f ( f  Fn  N  /\  A. n  e.  N  (
f `  n )  e.  { x  e.  A  |  ps } ) )
12 simprl 520 . . . . . 6  |-  ( (
ph  /\  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } ) )  -> 
f  Fn  N )
13 elrabi 2837 . . . . . . . 8  |-  ( ( f `  n )  e.  { x  e.  A  |  ps }  ->  ( f `  n
)  e.  A )
1413ralimi 2495 . . . . . . 7  |-  ( A. n  e.  N  (
f `  n )  e.  { x  e.  A  |  ps }  ->  A. n  e.  N  ( f `  n )  e.  A
)
1514ad2antll 482 . . . . . 6  |-  ( (
ph  /\  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } ) )  ->  A. n  e.  N  ( f `  n
)  e.  A )
16 nfcv 2281 . . . . . . 7  |-  F/_ n N
17 cc4f.a . . . . . . 7  |-  F/_ n A
18 nfcv 2281 . . . . . . 7  |-  F/_ n
f
1916, 17, 18ffnfvf 5579 . . . . . 6  |-  ( f : N --> A  <->  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  A
) )
2012, 15, 19sylanbrc 413 . . . . 5  |-  ( (
ph  /\  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } ) )  -> 
f : N --> A )
21 cc4f.3 . . . . . . . . 9  |-  ( x  =  ( f `  n )  ->  ( ps 
<->  ch ) )
2221elrab 2840 . . . . . . . 8  |-  ( ( f `  n )  e.  { x  e.  A  |  ps }  <->  ( ( f `  n
)  e.  A  /\  ch ) )
2322simprbi 273 . . . . . . 7  |-  ( ( f `  n )  e.  { x  e.  A  |  ps }  ->  ch )
2423ralimi 2495 . . . . . 6  |-  ( A. n  e.  N  (
f `  n )  e.  { x  e.  A  |  ps }  ->  A. n  e.  N  ch )
2524ad2antll 482 . . . . 5  |-  ( (
ph  /\  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } ) )  ->  A. n  e.  N  ch )
2620, 25jca 304 . . . 4  |-  ( (
ph  /\  ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } ) )  -> 
( f : N --> A  /\  A. n  e.  N  ch ) )
2726ex 114 . . 3  |-  ( ph  ->  ( ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  {
x  e.  A  |  ps } )  ->  (
f : N --> A  /\  A. n  e.  N  ch ) ) )
2827eximdv 1852 . 2  |-  ( ph  ->  ( E. f ( f  Fn  N  /\  A. n  e.  N  ( f `  n )  e.  { x  e.  A  |  ps }
)  ->  E. f
( f : N --> A  /\  A. n  e.  N  ch ) ) )
2911, 28mpd 13 1  |-  ( ph  ->  E. f ( f : N --> A  /\  A. n  e.  N  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   F/_wnfc 2268   A.wral 2416   E.wrex 2417   {crab 2420   _Vcvv 2686   class class class wbr 3929   omcom 4504    Fn wfn 5118   -->wf 5119   ` cfv 5123    ~~ cen 6632  CCHOICEwacc 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-2nd 6039  df-er 6429  df-en 6635  df-cc 7083
This theorem is referenced by:  cc4  7090
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