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Theorem fvreseq1 5989
Description: Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
Assertion
Ref Expression
fvreseq1  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  G  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hint:    A( x)

Proof of Theorem fvreseq1
StepHypRef Expression
1 fnresdm 5696 . . . . 5  |-  ( G  Fn  B  ->  ( G  |`  B )  =  G )
21ad2antlr 726 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( G  |`  B )  =  G )
32eqcomd 2465 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  G  =  ( G  |`  B ) )
43eqeq2d 2471 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  G  <->  ( F  |`  B )  =  ( G  |`  B )
) )
5 ssid 3518 . . 3  |-  B  C_  B
6 fvreseq0 5988 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( B  C_  A  /\  B  C_  B ) )  -> 
( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
75, 6mpanr2 684 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
84, 7bitrd 253 1  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  G  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   A.wral 2807    C_ wss 3471    |` cres 5010    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  symgextres  16576  sseqfres  28507
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