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Theorem funfvima3 5796
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 5675 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 ssel 3208 . . . . . 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 418 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  <. A ,  ( F `  A )
>.  e.  G )
5 sneq 3685 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
65imaeq2d 5049 . . . . . . 7  |-  ( x  =  A  ->  ( G " { x }
)  =  ( G
" { A }
) )
76eleq2d 2383 . . . . . 6  |-  ( x  =  A  ->  (
( F `  A
)  e.  ( G
" { x }
)  <->  ( F `  A )  e.  ( G " { A } ) ) )
8 opeq1 3833 . . . . . . 7  |-  ( x  =  A  ->  <. x ,  ( F `  A ) >.  =  <. A ,  ( F `  A ) >. )
98eleq1d 2382 . . . . . 6  |-  ( x  =  A  ->  ( <. x ,  ( F `
 A ) >.  e.  G  <->  <. A ,  ( F `  A )
>.  e.  G ) )
10 vex 2825 . . . . . . 7  |-  x  e. 
_V
11 fvex 5577 . . . . . . 7  |-  ( F `
 A )  e. 
_V
1210, 11elimasn 5075 . . . . . 6  |-  ( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `
 A ) >.  e.  G )
137, 9, 12vtoclbg 2878 . . . . 5  |-  ( A  e.  dom  F  -> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
1413ad2antll 709 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( ( F `
 A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
154, 14mpbird 223 . . 3  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( F `  A )  e.  ( G " { A } ) )
1615exp32 588 . 2  |-  ( F 
C_  G  ->  ( Fun  F  ->  ( A  e.  dom  F  ->  ( F `  A )  e.  ( G " { A } ) ) ) )
1716impcom 419 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    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186   {csn 3674   <.cop 3677   dom cdm 4726   "cima 4729   Fun wfun 5286   ` cfv 5292
This theorem is referenced by:  dfac3  7793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300
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