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Theorem funfvima3 5379
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 5266 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 ssel 2936 . . . . . 6  |-  ( F 
C_  G  ->  ( <. A ,  ( F `
 A ) >.  e.  F  ->  <. A , 
( F `  A
) >.  e.  G ) )
31, 2syl5 28 . . . . 5  |-  ( F 
C_  G  ->  (
( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `  A )
>.  e.  G ) )
43imp 115 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  <. A ,  ( F `  A )
>.  e.  G )
5 simpr 103 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  dom  F )
6 sneq 3383 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
76imaeq2d 4655 . . . . . . . . 9  |-  ( x  =  A  ->  ( G " { x }
)  =  ( G
" { A }
) )
87eleq2d 2107 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  A
)  e.  ( G
" { x }
)  <->  ( F `  A )  e.  ( G " { A } ) ) )
9 opeq1 3546 . . . . . . . . 9  |-  ( x  =  A  ->  <. x ,  ( F `  A ) >.  =  <. A ,  ( F `  A ) >. )
109eleq1d 2106 . . . . . . . 8  |-  ( x  =  A  ->  ( <. x ,  ( F `
 A ) >.  e.  G  <->  <. A ,  ( F `  A )
>.  e.  G ) )
118, 10bibi12d 224 . . . . . . 7  |-  ( x  =  A  ->  (
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G )  <-> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) ) )
1211adantl 262 . . . . . 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 2557 . . . . . . 7  |-  x  e. 
_V
14 funfvex 5179 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
15 elimasng 4680 . . . . . . 7  |-  ( ( x  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
1613, 14, 15sylancr 393 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
175, 12, 16vtocld 2603 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
1817adantl 262 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( ( F `
 A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
194, 18mpbird 156 . . 3  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( F `  A )  e.  ( G " { A } ) )
2019exp32 347 . 2  |-  ( F 
C_  G  ->  ( Fun  F  ->  ( A  e.  dom  F  ->  ( F `  A )  e.  ( G " { A } ) ) ) )
2120impcom 116 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 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2554    C_ wss 2914   {csn 3372   <.cop 3375   dom cdm 4332   "cima 4335   Fun wfun 4883   ` cfv 4889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-fv 4897
This theorem is referenced by: (None)
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