Theorem eqfnfv 5526

Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
eqfnfv	|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Distinct variable groups:   x, A   x, F   x, G

Proof of Theorem eqfnfv

Step	Hyp	Ref	Expression
1		dffn5im 5475	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		dffn5im 5475	. . 3 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	1, 2	eqeqan12d 2156	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) ) )
4		funfvex 5446	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
5	4	funfni 5231	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
6	5	ralrimiva 2508	. . . 4 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
7		mpteqb 5519	. . . 4 |- ( A. x e. A ( F ` x ) e. _V -> ( ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
8	6, 7	syl 14	. . 3 |- ( F Fn A -> ( ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
9	8	adantr 274	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
10	3, 9	bitrd 187	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   A.wral 2417   _Vcvv 2689   |-> cmpt 3997   Fn wfn 5126   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  eqfnfv2  5527  eqfnfvd  5529  eqfnfv2f  5530  fvreseq  5532  fneqeql  5536  fconst2g  5643  cocan1  5696  cocan2  5697  tfri3  6272  updjud  6975  ser0f  10319  prodf1f  11344  cnmpt11  12491  cnmpt21  12499