MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvreseq1 Structured version   Unicode version

Theorem fvreseq1 5803
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 5519 . . . . 5  |-  ( G  Fn  B  ->  ( G  |`  B )  =  G )
21ad2antlr 726 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( G  |`  B )  =  G )
32eqcomd 2447 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  G  =  ( G  |`  B ) )
43eqeq2d 2453 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  G  <->  ( F  |`  B )  =  ( G  |`  B )
) )
5 ssid 3374 . . 3  |-  B  C_  B
6 fvreseq0 5802 . . 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 1369   A.wral 2714    C_ wss 3327    |` cres 4841    Fn wfn 5412   ` cfv 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-fv 5425
This theorem is referenced by:  symgextres  15929  sseqfres  26775
  Copyright terms: Public domain W3C validator