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Theorem funfvima3 6066
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 5927 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 ssel 3461 . . . . . 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 429 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  <. A ,  ( F `  A )
>.  e.  G )
5 sneq 3998 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
65imaeq2d 5280 . . . . . . 7  |-  ( x  =  A  ->  ( G " { x }
)  =  ( G
" { A }
) )
76eleq2d 2524 . . . . . 6  |-  ( x  =  A  ->  (
( F `  A
)  e.  ( G
" { x }
)  <->  ( F `  A )  e.  ( G " { A } ) ) )
8 opeq1 4170 . . . . . . 7  |-  ( x  =  A  ->  <. x ,  ( F `  A ) >.  =  <. A ,  ( F `  A ) >. )
98eleq1d 2523 . . . . . 6  |-  ( x  =  A  ->  ( <. x ,  ( F `
 A ) >.  e.  G  <->  <. A ,  ( F `  A )
>.  e.  G ) )
10 vex 3081 . . . . . . 7  |-  x  e. 
_V
11 fvex 5812 . . . . . . 7  |-  ( F `
 A )  e. 
_V
1210, 11elimasn 5305 . . . . . 6  |-  ( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `
 A ) >.  e.  G )
137, 9, 12vtoclbg 3137 . . . . 5  |-  ( A  e.  dom  F  -> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
1413ad2antll 728 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( ( F `
 A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
154, 14mpbird 232 . . 3  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( F `  A )  e.  ( G " { A } ) )
1615exp32 605 . 2  |-  ( F 
C_  G  ->  ( Fun  F  ->  ( A  e.  dom  F  ->  ( F `  A )  e.  ( G " { A } ) ) ) )
1716impcom 430 1  |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F  -> 
( F `  A
)  e.  ( G
" { A }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3439   {csn 3988   <.cop 3994   dom cdm 4951   "cima 4954   Fun wfun 5523   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537
This theorem is referenced by:  dfac3  8406
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