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Theorem eqfnfv2 5983
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 5213 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
2 fndm 5686 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
3 fndm 5686 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3eqeqan12d 2480 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  =  dom  G  <->  A  =  B ) )
51, 4syl5ib 219 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  ->  A  =  B ) )
65pm4.71rd 635 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  F  =  G ) ) )
7 fneq2 5676 . . . . . 6  |-  ( A  =  B  ->  ( G  Fn  A  <->  G  Fn  B ) )
87biimparc 487 . . . . 5  |-  ( ( G  Fn  B  /\  A  =  B )  ->  G  Fn  A )
9 eqfnfv 5982 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9sylan2 474 . . . 4  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  A  =  B
) )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1110anassrs 648 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  A  =  B )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1211pm5.32da 641 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  =  B  /\  F  =  G )  <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
136, 12bitrd 253 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   A.wral 2807   dom cdm 5008    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:  eqfnfv3  5984  eqfunfv  5987  eqfnov  6407  2ffzeq  11820  eqwrd  12590  soseq  29551  wfr3g  29559  frr3g  29603  nodenselem4  29661  sdclem2  30440  2ffzoeq  32605
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