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Theorem funfvima3 5700
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 5579 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 ssel 3122 . . . . . 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 123 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  <. A ,  ( F `  A )
>.  e.  G )
5 simpr 109 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  dom  F )
6 sneq 3571 . . . . . . . . . 10  |-  ( x  =  A  ->  { x }  =  { A } )
76imaeq2d 4928 . . . . . . . . 9  |-  ( x  =  A  ->  ( G " { x }
)  =  ( G
" { A }
) )
87eleq2d 2227 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  A
)  e.  ( G
" { x }
)  <->  ( F `  A )  e.  ( G " { A } ) ) )
9 opeq1 3741 . . . . . . . . 9  |-  ( x  =  A  ->  <. x ,  ( F `  A ) >.  =  <. A ,  ( F `  A ) >. )
109eleq1d 2226 . . . . . . . 8  |-  ( x  =  A  ->  ( <. x ,  ( F `
 A ) >.  e.  G  <->  <. A ,  ( F `  A )
>.  e.  G ) )
118, 10bibi12d 234 . . . . . . 7  |-  ( x  =  A  ->  (
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G )  <-> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) ) )
1211adantl 275 . . . . . 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 2715 . . . . . . 7  |-  x  e. 
_V
14 funfvex 5485 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
15 elimasng 4954 . . . . . . 7  |-  ( ( x  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
1613, 14, 15sylancr 411 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { x } )  <->  <. x ,  ( F `  A
) >.  e.  G ) )
175, 12, 16vtocld 2764 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
1817adantl 275 . . . 4  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( ( F `
 A )  e.  ( G " { A } )  <->  <. A , 
( F `  A
) >.  e.  G ) )
194, 18mpbird 166 . . 3  |-  ( ( F  C_  G  /\  ( Fun  F  /\  A  e.  dom  F ) )  ->  ( F `  A )  e.  ( G " { A } ) )
2019exp32 363 . 2  |-  ( F 
C_  G  ->  ( Fun  F  ->  ( A  e.  dom  F  ->  ( F `  A )  e.  ( G " { A } ) ) ) )
2120impcom 124 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 103    <-> wb 104    = wceq 1335    e. wcel 2128   _Vcvv 2712    C_ wss 3102   {csn 3560   <.cop 3563   dom cdm 4586   "cima 4589   Fun wfun 5164   ` cfv 5170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by: (None)
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