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Theorem fveqres 5898
Description: Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
fveqres  |-  ( ( F `  A )  =  ( G `  A )  ->  (
( F  |`  B ) `
 A )  =  ( ( G  |`  B ) `  A
) )

Proof of Theorem fveqres
StepHypRef Expression
1 fvres 5878 . . . 4  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
2 fvres 5878 . . . 4  |-  ( A  e.  B  ->  (
( G  |`  B ) `
 A )  =  ( G `  A
) )
31, 2eqeq12d 2489 . . 3  |-  ( A  e.  B  ->  (
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A )  <->  ( F `  A )  =  ( G `  A ) ) )
43biimprd 223 . 2  |-  ( A  e.  B  ->  (
( F `  A
)  =  ( G `
 A )  -> 
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) ) )
5 nfvres 5894 . . . 4  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
6 nfvres 5894 . . . 4  |-  ( -.  A  e.  B  -> 
( ( G  |`  B ) `  A
)  =  (/) )
75, 6eqtr4d 2511 . . 3  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) )
87a1d 25 . 2  |-  ( -.  A  e.  B  -> 
( ( F `  A )  =  ( G `  A )  ->  ( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) ) )
94, 8pm2.61i 164 1  |-  ( ( F `  A )  =  ( G `  A )  ->  (
( F  |`  B ) `
 A )  =  ( ( G  |`  B ) `  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   (/)c0 3785    |` cres 5001   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-res 5011  df-iota 5549  df-fv 5594
This theorem is referenced by:  fvresex  6754
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