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Theorem eqfnfv2 5623
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 4879 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
2 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
3 fndm 5343 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3eqeqan12d 2298 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  =  dom  G  <->  A  =  B ) )
51, 4syl5ib 210 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  ->  A  =  B ) )
65pm4.71rd 616 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  F  =  G ) ) )
7 fneq2 5334 . . . . . 6  |-  ( A  =  B  ->  ( G  Fn  A  <->  G  Fn  B ) )
87biimparc 473 . . . . 5  |-  ( ( G  Fn  B  /\  A  =  B )  ->  G  Fn  A )
9 eqfnfv 5622 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9sylan2 460 . . . 4  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  A  =  B
) )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1110anassrs 629 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  A  =  B )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1211pm5.32da 622 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  =  B  /\  F  =  G )  <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
136, 12bitrd 244 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 176    /\ wa 358    = wceq 1623   A.wral 2543   dom cdm 4689    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  eqfnfv3  5624  eqfunfv  5627  eqfnov  5950  soseq  24254  wfr3g  24255  frr3g  24280  nodenselem4  24338  eqfnung2  25118  sdclem2  26452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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