MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fveqres Structured version   Unicode version

Theorem fveqres 5764
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 5745 . . . 4  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
2 fvres 5745 . . . 4  |-  ( A  e.  B  ->  (
( G  |`  B ) `
 A )  =  ( G `  A
) )
31, 2eqeq12d 2450 . . 3  |-  ( A  e.  B  ->  (
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A )  <->  ( F `  A )  =  ( G `  A ) ) )
43biimprd 215 . 2  |-  ( A  e.  B  ->  (
( F `  A
)  =  ( G `
 A )  -> 
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) ) )
5 nfvres 5760 . . . 4  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
6 nfvres 5760 . . . 4  |-  ( -.  A  e.  B  -> 
( ( G  |`  B ) `  A
)  =  (/) )
75, 6eqtr4d 2471 . . 3  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) )
87a1d 23 . 2  |-  ( -.  A  e.  B  -> 
( ( F `  A )  =  ( G `  A )  ->  ( ( F  |`  B ) `  A
)  =  ( ( G  |`  B ) `  A ) ) )
94, 8pm2.61i 158 1  |-  ( ( F `  A )  =  ( G `  A )  ->  (
( F  |`  B ) `
 A )  =  ( ( G  |`  B ) `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   (/)c0 3628    |` cres 4880   ` cfv 5454
This theorem is referenced by:  fvresex  5982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-res 4890  df-iota 5418  df-fv 5462
  Copyright terms: Public domain W3C validator