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Theorem ccfunen 7226
Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
Hypotheses
Ref Expression
ccfunen.cc  |-  ( ph  -> CCHOICE )
ccfunen.a  |-  ( ph  ->  A  ~~  om )
ccfunen.m  |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )
Assertion
Ref Expression
ccfunen  |-  ( ph  ->  E. f ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  e.  x ) )
Distinct variable groups:    A, f, x    ph, f, x    x, w
Allowed substitution hints:    ph( w)    A( w)

Proof of Theorem ccfunen
Dummy variables  u  v  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ccfunen.a . . . . . 6  |-  ( ph  ->  A  ~~  om )
2 encv 6724 . . . . . 6  |-  ( A 
~~  om  ->  ( A  e.  _V  /\  om  e.  _V ) )
31, 2syl 14 . . . . 5  |-  ( ph  ->  ( A  e.  _V  /\ 
om  e.  _V )
)
43simpld 111 . . . 4  |-  ( ph  ->  A  e.  _V )
5 abid2 2291 . . . . . 6  |-  { v  |  v  e.  u }  =  u
6 vex 2733 . . . . . 6  |-  u  e. 
_V
75, 6eqeltri 2243 . . . . 5  |-  { v  |  v  e.  u }  e.  _V
87a1i 9 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  { v  |  v  e.  u }  e.  _V )
94, 8opabex3d 6100 . . 3  |-  ( ph  ->  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V )
10 ccfunen.cc . . . 4  |-  ( ph  -> CCHOICE )
11 df-cc 7225 . . . 4  |-  (CCHOICE  <->  A. y
( dom  y  ~~  om 
->  E. f ( f 
C_  y  /\  f  Fn  dom  y ) ) )
1210, 11sylib 121 . . 3  |-  ( ph  ->  A. y ( dom  y  ~~  om  ->  E. f ( f  C_  y  /\  f  Fn  dom  y ) ) )
13 ccfunen.m . . . . . 6  |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )
14 elequ2 2146 . . . . . . . . 9  |-  ( x  =  u  ->  (
w  e.  x  <->  w  e.  u ) )
1514exbidv 1818 . . . . . . . 8  |-  ( x  =  u  ->  ( E. w  w  e.  x 
<->  E. w  w  e.  u ) )
1615cbvralv 2696 . . . . . . 7  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. w  w  e.  u
)
17 elequ1 2145 . . . . . . . . 9  |-  ( w  =  v  ->  (
w  e.  u  <->  v  e.  u ) )
1817cbvexv 1911 . . . . . . . 8  |-  ( E. w  w  e.  u  <->  E. v  v  e.  u
)
1918ralbii 2476 . . . . . . 7  |-  ( A. u  e.  A  E. w  w  e.  u  <->  A. u  e.  A  E. v  v  e.  u
)
2016, 19bitri 183 . . . . . 6  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. v  v  e.  u
)
2113, 20sylib 121 . . . . 5  |-  ( ph  ->  A. u  e.  A  E. v  v  e.  u )
22 dmopab3 4824 . . . . 5  |-  ( A. u  e.  A  E. v  v  e.  u  <->  dom 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  =  A )
2321, 22sylib 121 . . . 4  |-  ( ph  ->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  =  A )
2423, 1eqbrtrd 4011 . . 3  |-  ( ph  ->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om )
25 dmeq 4811 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  dom  y  =  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } )
2625breq1d 3999 . . . . 5  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( dom  y  ~~  om  <->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om ) )
27 sseq2 3171 . . . . . . 7  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  C_  y 
<->  f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )
2825fneq2d 5289 . . . . . . 7  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  Fn 
dom  y  <->  f  Fn  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } ) )
2927, 28anbi12d 470 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( ( f 
C_  y  /\  f  Fn  dom  y )  <->  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) ) )
3029exbidv 1818 . . . . 5  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( E. f
( f  C_  y  /\  f  Fn  dom  y )  <->  E. f
( f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) ) )
3126, 30imbi12d 233 . . . 4  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( ( dom  y  ~~  om  ->  E. f ( f  C_  y  /\  f  Fn  dom  y ) )  <->  ( dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om 
->  E. f ( f 
C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) ) ) )
3231spcgv 2817 . . 3  |-  ( {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V  ->  ( A. y
( dom  y  ~~  om 
->  E. f ( f 
C_  y  /\  f  Fn  dom  y ) )  ->  ( dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om 
->  E. f ( f 
C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) ) ) )
339, 12, 24, 32syl3c 63 . 2  |-  ( ph  ->  E. f ( f 
C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )
34 simprr 527 . . . . . 6  |-  ( (
ph  /\  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  -> 
f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } )
3523fneq2d 5289 . . . . . . 7  |-  ( ph  ->  ( f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  <->  f  Fn  A ) )
3635adantr 274 . . . . . 6  |-  ( (
ph  /\  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  -> 
( f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  <->  f  Fn  A ) )
3734, 36mpbid 146 . . . . 5  |-  ( (
ph  /\  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  -> 
f  Fn  A )
38 simplrl 530 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  /\  x  e.  A )  ->  f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } )
39 fnopfv 5626 . . . . . . . . . 10  |-  ( ( f  Fn  A  /\  x  e.  A )  -> 
<. x ,  ( f `
 x ) >.  e.  f )
4037, 39sylan 281 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  /\  x  e.  A )  ->  <. x ,  ( f `  x ) >.  e.  f )
4138, 40sseldd 3148 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  /\  x  e.  A )  ->  <. x ,  ( f `  x ) >.  e.  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } )
42 vex 2733 . . . . . . . . 9  |-  x  e. 
_V
43 vex 2733 . . . . . . . . . 10  |-  f  e. 
_V
4443, 42fvex 5516 . . . . . . . . 9  |-  ( f `
 x )  e. 
_V
45 eleq1 2233 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  A  <->  x  e.  A ) )
46 elequ2 2146 . . . . . . . . . 10  |-  ( u  =  x  ->  (
v  e.  u  <->  v  e.  x ) )
4745, 46anbi12d 470 . . . . . . . . 9  |-  ( u  =  x  ->  (
( u  e.  A  /\  v  e.  u
)  <->  ( x  e.  A  /\  v  e.  x ) ) )
48 eleq1 2233 . . . . . . . . . 10  |-  ( v  =  ( f `  x )  ->  (
v  e.  x  <->  ( f `  x )  e.  x
) )
4948anbi2d 461 . . . . . . . . 9  |-  ( v  =  ( f `  x )  ->  (
( x  e.  A  /\  v  e.  x
)  <->  ( x  e.  A  /\  ( f `
 x )  e.  x ) ) )
5042, 44, 47, 49opelopab 4256 . . . . . . . 8  |-  ( <.
x ,  ( f `
 x ) >.  e.  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  <->  ( x  e.  A  /\  (
f `  x )  e.  x ) )
5141, 50sylib 121 . . . . . . 7  |-  ( ( ( ph  /\  (
f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  /\  x  e.  A )  ->  (
x  e.  A  /\  ( f `  x
)  e.  x ) )
5251simprd 113 . . . . . 6  |-  ( ( ( ph  /\  (
f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  /\  x  e.  A )  ->  (
f `  x )  e.  x )
5352ralrimiva 2543 . . . . 5  |-  ( (
ph  /\  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  ->  A. x  e.  A  ( f `  x
)  e.  x )
5437, 53jca 304 . . . 4  |-  ( (
ph  /\  ( f  C_ 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )  -> 
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  x ) )
5554ex 114 . . 3  |-  ( ph  ->  ( ( f  C_  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } )  ->  (
f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  x ) ) )
5655eximdv 1873 . 2  |-  ( ph  ->  ( E. f ( f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  /\  f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } )  ->  E. f ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  e.  x ) ) )
5733, 56mpd 13 1  |-  ( ph  ->  E. f ( f  Fn  A  /\  A. x  e.  A  (
f `  x )  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   A.wral 2448   _Vcvv 2730    C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049   omcom 4574   dom cdm 4611    Fn wfn 5193   ` cfv 5198    ~~ cen 6716  CCHOICEwacc 7224
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-en 6719  df-cc 7225
This theorem is referenced by:  cc1  7227  cc2lem  7228
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