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Theorem ccfunen 7184
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 6691 . . . . . 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 2278 . . . . . 6  |-  { v  |  v  e.  u }  =  u
6 vex 2715 . . . . . 6  |-  u  e. 
_V
75, 6eqeltri 2230 . . . . 5  |-  { v  |  v  e.  u }  e.  _V
87a1i 9 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  { v  |  v  e.  u }  e.  _V )
94, 8opabex3d 6069 . . 3  |-  ( ph  ->  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V )
10 ccfunen.cc . . . 4  |-  ( ph  -> CCHOICE )
11 df-cc 7183 . . . 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 2133 . . . . . . . . 9  |-  ( x  =  u  ->  (
w  e.  x  <->  w  e.  u ) )
1514exbidv 1805 . . . . . . . 8  |-  ( x  =  u  ->  ( E. w  w  e.  x 
<->  E. w  w  e.  u ) )
1615cbvralv 2680 . . . . . . 7  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. w  w  e.  u
)
17 elequ1 2132 . . . . . . . . 9  |-  ( w  =  v  ->  (
w  e.  u  <->  v  e.  u ) )
1817cbvexv 1898 . . . . . . . 8  |-  ( E. w  w  e.  u  <->  E. v  v  e.  u
)
1918ralbii 2463 . . . . . . 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 4799 . . . . 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 3986 . . 3  |-  ( ph  ->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om )
25 dmeq 4786 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  dom  y  =  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } )
2625breq1d 3975 . . . . 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 3152 . . . . . . 7  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  C_  y 
<->  f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )
2825fneq2d 5261 . . . . . . 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 465 . . . . . 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 1805 . . . . 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 2799 . . 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 522 . . . . . 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 5261 . . . . . . 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 525 . . . . . . . . 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 5597 . . . . . . . . . 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 3129 . . . . . . . 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 2715 . . . . . . . . 9  |-  x  e. 
_V
43 vex 2715 . . . . . . . . . 10  |-  f  e. 
_V
4443, 42fvex 5488 . . . . . . . . 9  |-  ( f `
 x )  e. 
_V
45 eleq1 2220 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  A  <->  x  e.  A ) )
46 elequ2 2133 . . . . . . . . . 10  |-  ( u  =  x  ->  (
v  e.  u  <->  v  e.  x ) )
4745, 46anbi12d 465 . . . . . . . . 9  |-  ( u  =  x  ->  (
( u  e.  A  /\  v  e.  u
)  <->  ( x  e.  A  /\  v  e.  x ) ) )
48 eleq1 2220 . . . . . . . . . 10  |-  ( v  =  ( f `  x )  ->  (
v  e.  x  <->  ( f `  x )  e.  x
) )
4948anbi2d 460 . . . . . . . . 9  |-  ( v  =  ( f `  x )  ->  (
( x  e.  A  /\  v  e.  x
)  <->  ( x  e.  A  /\  ( f `
 x )  e.  x ) ) )
5042, 44, 47, 49opelopab 4231 . . . . . . . 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 2530 . . . . 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 1860 . 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 1333    = wceq 1335   E.wex 1472    e. wcel 2128   {cab 2143   A.wral 2435   _Vcvv 2712    C_ wss 3102   <.cop 3563   class class class wbr 3965   {copab 4024   omcom 4549   dom cdm 4586    Fn wfn 5165   ` cfv 5170    ~~ cen 6683  CCHOICEwacc 7182
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-en 6686  df-cc 7183
This theorem is referenced by:  cc1  7185  cc2lem  7186
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