ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ccfunen Unicode version

Theorem ccfunen 7072
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 6633 . . . . . 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 2258 . . . . . 6  |-  { v  |  v  e.  u }  =  u
6 vex 2684 . . . . . 6  |-  u  e. 
_V
75, 6eqeltri 2210 . . . . 5  |-  { v  |  v  e.  u }  e.  _V
87a1i 9 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  { v  |  v  e.  u }  e.  _V )
94, 8opabex3d 6012 . . 3  |-  ( ph  ->  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V )
10 ccfunen.cc . . . 4  |-  ( ph  -> CCHOICE )
11 df-cc 7071 . . . 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 1691 . . . . . . . . 9  |-  ( x  =  u  ->  (
w  e.  x  <->  w  e.  u ) )
1514exbidv 1797 . . . . . . . 8  |-  ( x  =  u  ->  ( E. w  w  e.  x 
<->  E. w  w  e.  u ) )
1615cbvralv 2652 . . . . . . 7  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. w  w  e.  u
)
17 elequ1 1690 . . . . . . . . 9  |-  ( w  =  v  ->  (
w  e.  u  <->  v  e.  u ) )
1817cbvexv 1890 . . . . . . . 8  |-  ( E. w  w  e.  u  <->  E. v  v  e.  u
)
1918ralbii 2439 . . . . . . 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 4747 . . . . 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 3945 . . 3  |-  ( ph  ->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ~~  om )
25 dmeq 4734 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  dom  y  =  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } )
2625breq1d 3934 . . . . 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 3116 . . . . . . 7  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  C_  y 
<->  f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )
2825fneq2d 5209 . . . . . . 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 464 . . . . . 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 1797 . . . . 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 2768 . . 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 521 . . . . . 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 5209 . . . . . . 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 524 . . . . . . . . 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 5543 . . . . . . . . . 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 3093 . . . . . . . 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 2684 . . . . . . . . 9  |-  x  e. 
_V
43 vex 2684 . . . . . . . . . 10  |-  f  e. 
_V
4443, 42fvex 5434 . . . . . . . . 9  |-  ( f `
 x )  e. 
_V
45 eleq1 2200 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  A  <->  x  e.  A ) )
46 elequ2 1691 . . . . . . . . . 10  |-  ( u  =  x  ->  (
v  e.  u  <->  v  e.  x ) )
4745, 46anbi12d 464 . . . . . . . . 9  |-  ( u  =  x  ->  (
( u  e.  A  /\  v  e.  u
)  <->  ( x  e.  A  /\  v  e.  x ) ) )
48 eleq1 2200 . . . . . . . . . 10  |-  ( v  =  ( f `  x )  ->  (
v  e.  x  <->  ( f `  x )  e.  x
) )
4948anbi2d 459 . . . . . . . . 9  |-  ( v  =  ( f `  x )  ->  (
( x  e.  A  /\  v  e.  x
)  <->  ( x  e.  A  /\  ( f `
 x )  e.  x ) ) )
5042, 44, 47, 49opelopab 4188 . . . . . . . 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 2503 . . . . 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 1852 . 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 1329    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2123   A.wral 2414   _Vcvv 2681    C_ wss 3066   <.cop 3525   class class class wbr 3924   {copab 3983   omcom 4499   dom cdm 4534    Fn wfn 5113   ` cfv 5118    ~~ cen 6625  CCHOICEwacc 7070
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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-en 6628  df-cc 7071
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator