MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfnfv2 Unicode version

Theorem eqfnfv2 5814
Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
Distinct variable groups:    x, A    x, F    x, G
Allowed substitution hint:    B( x)

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 5056 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
2 fndm 5530 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
3 fndm 5530 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3eqeqan12d 2445 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  =  dom  G  <->  A  =  B ) )
51, 4syl5ib 211 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  ->  A  =  B ) )
65pm4.71rd 617 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  F  =  G ) ) )
7 fneq2 5521 . . . . . 6  |-  ( A  =  B  ->  ( G  Fn  A  <->  G  Fn  B ) )
87biimparc 474 . . . . 5  |-  ( ( G  Fn  B  /\  A  =  B )  ->  G  Fn  A )
9 eqfnfv 5813 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9sylan2 461 . . . 4  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  A  =  B
) )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1110anassrs 630 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  A  =  B )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1211pm5.32da 623 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  =  B  /\  F  =  G )  <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
136, 12bitrd 245 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652   A.wral 2692   dom cdm 4864    Fn wfn 5435   ` cfv 5440
This theorem is referenced by:  eqfnfv3  5815  eqfunfv  5818  eqfnov  6162  soseq  25504  wfr3g  25505  frr3g  25530  nodenselem4  25588  sdclem2  26378
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390
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 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-fv 5448
  Copyright terms: Public domain W3C validator