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Theorem funfvima3 5528
Description: A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
Assertion
Ref Expression
funfvima3  |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F  -> 
( F `  A
)  e.  ( G
" { A }
) ) )

Proof of Theorem funfvima3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funfvop 5411 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 ssel 3019 . . . . . 6  |-  ( F 
C_  G  ->  ( <. A ,  ( F `
 A ) >.  e.  F  ->  <. A , 
( F `  A
) >.  e.  G ) )
31, 2syl5 32 . . . . 5  |-  ( F 
C_  G  ->  (
( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `  A )
>.  e.  G ) )
43imp 122 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  <. A ,  ( F `  A )
>.  e.  G )
5 simpr 108 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  dom  F )
6 sneq 3457 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
76imaeq2d 4774 . . . . . . . . 9  |-  ( x  =  A  ->  ( G " { x }
)  =  ( G
" { A }
) )
87eleq2d 2157 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  A
)  e.  ( G
" { x }
)  <->  ( F `  A )  e.  ( G " { A } ) ) )
9 opeq1 3622 . . . . . . . . 9  |-  ( x  =  A  ->  <. x ,  ( F `  A ) >.  =  <. A ,  ( F `  A ) >. )
109eleq1d 2156 . . . . . . . 8  |-  ( x  =  A  ->  ( <. x ,  ( F `
 A ) >.  e.  G  <->  <. A ,  ( F `  A )
>.  e.  G ) )
118, 10bibi12d 233 . . . . . . 7  |-  ( x  =  A  ->  (
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G )  <-> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) ) )
1211adantl 271 . . . . . 6  |-  ( ( ( Fun  F  /\  A  e.  dom  F )  /\  x  =  A )  ->  ( (
( F `  A
)  e.  ( G
" { x }
)  <->  <. x ,  ( F `  A )
>.  e.  G )  <->  ( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) ) )
13 vex 2622 . . . . . . 7  |-  x  e. 
_V
14 funfvex 5322 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
15 elimasng 4800 . . . . . . 7  |-  ( ( x  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
1613, 14, 15sylancr 405 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
175, 12, 16vtocld 2671 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
1817adantl 271 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( ( F `
 A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
194, 18mpbird 165 . . 3  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( F `  A )  e.  ( G " { A } ) )
2019exp32 357 . 2  |-  ( F 
C_  G  ->  ( Fun  F  ->  ( A  e.  dom  F  ->  ( F `  A )  e.  ( G " { A } ) ) ) )
2120impcom 123 1  |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F  -> 
( F `  A
)  e.  ( G
" { A }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2999   {csn 3446   <.cop 3449   dom cdm 4438   "cima 4441   Fun wfun 5009   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by: (None)
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