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Theorem fveqres 5882
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 5862 . . . 4  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
2 fvres 5862 . . . 4  |-  ( A  e.  B  ->  (
( G  |`  B ) `
 A )  =  ( G `  A
) )
31, 2eqeq12d 2424 . . 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 5878 . . . 4  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
6 nfvres 5878 . . . 4  |-  ( -.  A  e.  B  -> 
( ( G  |`  B ) `  A
)  =  (/) )
75, 6eqtr4d 2446 . . 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 1405    e. wcel 1842   (/)c0 3737    |` cres 4824   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4828  df-dm 4832  df-res 4834  df-iota 5532  df-fv 5576
This theorem is referenced by:  fvresex  6756
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