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Theorem acfun 7200
Description: A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
Hypotheses
Ref Expression
acfun.ac  |-  ( ph  -> CHOICE )
acfun.a  |-  ( ph  ->  A  e.  V )
acfun.m  |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )
Assertion
Ref Expression
acfun  |-  ( 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)    V( x, w, f)

Proof of Theorem acfun
Dummy variables  u  v  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 acfun.a . . . . 5  |-  ( ph  ->  A  e.  V )
21elexd 2750 . . . 4  |-  ( ph  ->  A  e.  _V )
3 abid2 2298 . . . . . 6  |-  { v  |  v  e.  u }  =  u
4 vex 2740 . . . . . 6  |-  u  e. 
_V
53, 4eqeltri 2250 . . . . 5  |-  { v  |  v  e.  u }  e.  _V
65a1i 9 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  { v  |  v  e.  u }  e.  _V )
72, 6opabex3d 6116 . . 3  |-  ( ph  ->  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V )
8 acfun.ac . . . 4  |-  ( ph  -> CHOICE )
9 df-ac 7199 . . . 4  |-  (CHOICE  <->  A. y E. f ( f  C_  y  /\  f  Fn  dom  y ) )
108, 9sylib 122 . . 3  |-  ( ph  ->  A. y E. f
( f  C_  y  /\  f  Fn  dom  y ) )
11 sseq2 3179 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  C_  y 
<->  f  C_  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) } ) )
12 dmeq 4823 . . . . . . 7  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  dom  y  =  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } )
1312fneq2d 5303 . . . . . 6  |-  ( y  =  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  ->  ( f  Fn 
dom  y  <->  f  Fn  dom  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) } ) )
1411, 13anbi12d 473 . . . . 5  |-  ( 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 ) } ) ) )
1514exbidv 1825 . . . 4  |-  ( 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 ) } ) ) )
1615spcgv 2824 . . 3  |-  ( {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  e.  _V  ->  ( A. y 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 ) } ) ) )
177, 10, 16sylc 62 . 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 ) } ) )
18 simprr 531 . . . . . 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 ) } )
19 acfun.m . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )
20 elequ2 2153 . . . . . . . . . . . . 13  |-  ( x  =  u  ->  (
w  e.  x  <->  w  e.  u ) )
2120exbidv 1825 . . . . . . . . . . . 12  |-  ( x  =  u  ->  ( E. w  w  e.  x 
<->  E. w  w  e.  u ) )
2221cbvralv 2703 . . . . . . . . . . 11  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. w  w  e.  u
)
23 elequ1 2152 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
w  e.  u  <->  v  e.  u ) )
2423cbvexv 1918 . . . . . . . . . . . 12  |-  ( E. w  w  e.  u  <->  E. v  v  e.  u
)
2524ralbii 2483 . . . . . . . . . . 11  |-  ( A. u  e.  A  E. w  w  e.  u  <->  A. u  e.  A  E. v  v  e.  u
)
2622, 25bitri 184 . . . . . . . . . 10  |-  ( A. x  e.  A  E. w  w  e.  x  <->  A. u  e.  A  E. v  v  e.  u
)
2719, 26sylib 122 . . . . . . . . 9  |-  ( ph  ->  A. u  e.  A  E. v  v  e.  u )
28 dmopab3 4836 . . . . . . . . 9  |-  ( A. u  e.  A  E. v  v  e.  u  <->  dom 
{ <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  =  A )
2927, 28sylib 122 . . . . . . . 8  |-  ( ph  ->  dom  { <. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  =  A )
3029fneq2d 5303 . . . . . . 7  |-  ( ph  ->  ( f  Fn  dom  {
<. u ,  v >.  |  ( u  e.  A  /\  v  e.  u ) }  <->  f  Fn  A ) )
3130adantr 276 . . . . . 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 ) )
3218, 31mpbid 147 . . . . 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 )
33 simplrl 535 . . . . . . . . 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 ) } )
34 fnopfv 5642 . . . . . . . . . 10  |-  ( ( f  Fn  A  /\  x  e.  A )  -> 
<. x ,  ( f `
 x ) >.  e.  f )
3532, 34sylan 283 . . . . . . . . 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 )
3633, 35sseldd 3156 . . . . . . . 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 ) } )
37 vex 2740 . . . . . . . . 9  |-  x  e. 
_V
38 vex 2740 . . . . . . . . . 10  |-  f  e. 
_V
3938, 37fvex 5531 . . . . . . . . 9  |-  ( f `
 x )  e. 
_V
40 eleq1 2240 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  A  <->  x  e.  A ) )
41 elequ2 2153 . . . . . . . . . 10  |-  ( u  =  x  ->  (
v  e.  u  <->  v  e.  x ) )
4240, 41anbi12d 473 . . . . . . . . 9  |-  ( u  =  x  ->  (
( u  e.  A  /\  v  e.  u
)  <->  ( x  e.  A  /\  v  e.  x ) ) )
43 eleq1 2240 . . . . . . . . . 10  |-  ( v  =  ( f `  x )  ->  (
v  e.  x  <->  ( f `  x )  e.  x
) )
4443anbi2d 464 . . . . . . . . 9  |-  ( v  =  ( f `  x )  ->  (
( x  e.  A  /\  v  e.  x
)  <->  ( x  e.  A  /\  ( f `
 x )  e.  x ) ) )
4537, 39, 42, 44opelopab 4268 . . . . . . . 8  |-  ( <.
x ,  ( f `
 x ) >.  e.  { <. u ,  v
>.  |  ( u  e.  A  /\  v  e.  u ) }  <->  ( x  e.  A  /\  (
f `  x )  e.  x ) )
4636, 45sylib 122 . . . . . . 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 ) )
4746simprd 114 . . . . . 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 )
4847ralrimiva 2550 . . . . 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 )
4932, 48jca 306 . . . 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 ) )
5049ex 115 . . 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 ) ) )
5150eximdv 1880 . 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 ) ) )
5217, 51mpd 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 104    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   _Vcvv 2737    C_ wss 3129   <.cop 3594   {copab 4060   dom cdm 4623    Fn wfn 5207   ` cfv 5212  CHOICEwac 7198
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ac 7199
This theorem is referenced by:  exmidaclem  7201
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