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Theorem eqfnfv2 5648
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 4856 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
2 fndm 5345 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
3 fndm 5345 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3eqeqan12d 2209 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  =  dom  G  <->  A  =  B ) )
51, 4imbitrid 154 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  ->  A  =  B ) )
65pm4.71rd 394 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  F  =  G ) ) )
7 fneq2 5335 . . . . . 6  |-  ( A  =  B  ->  ( G  Fn  A  <->  G  Fn  B ) )
87biimparc 299 . . . . 5  |-  ( ( G  Fn  B  /\  A  =  B )  ->  G  Fn  A )
9 eqfnfv 5647 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9sylan2 286 . . . 4  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  A  =  B
) )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1110anassrs 400 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  A  =  B )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
1211pm5.32da 452 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  =  B  /\  F  =  G )  <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
136, 12bitrd 188 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    /\ wa 104    <-> wb 105    = wceq 1364   A.wral 2472   dom cdm 4655    Fn wfn 5241   ` cfv 5246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254
This theorem is referenced by:  eqfnfv3  5649  eqfunfv  5652  eqfnov  6017  2ffzeq  10197  eqwrd  10944
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