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Theorem eqfnfv2 5584
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 4878 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
2 fndm 5308 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
3 fndm 5308 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3eqeqan12d 2299 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  =  dom  G  <->  A  =  B ) )
51, 4syl5ib 212 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  ->  A  =  B ) )
65pm4.71rd 619 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  F  =  G ) ) )
7 fneq2 5299 . . . . . 6  |-  ( A  =  B  ->  ( G  Fn  A  <->  G  Fn  B ) )
87biimparc 475 . . . . 5  |-  ( ( G  Fn  B  /\  A  =  B )  ->  G  Fn  A )
9 eqfnfv 5583 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9sylan2 462 . . . 4  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  A  =  B
) )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1110anassrs 632 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  A  =  B )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1211pm5.32da 625 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  =  B  /\  F  =  G )  <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
136, 12bitrd 246 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 6    <-> wb 178    /\ wa 360    = wceq 1628   A.wral 2544   dom cdm 4688    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  eqfnfv3  5585  eqfunfv  5588  eqfnov  5911  soseq  23655  wfr3g  23656  frr3g  23681  axdenselem4  23739  eqfnung2  24517  sdclem2  25851
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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