Theorem eqfnfv 5659

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 5606	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		dffn5im 5606	. . 3 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	1, 2	eqeqan12d 2212	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) ) )
4		funfvex 5575	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
5	4	funfni 5358	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
6	5	ralrimiva 2570	. . . 4 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
7		mpteqb 5652	. . . 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 276	. 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 188	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   |-> cmpt 4094   Fn wfn 5253   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  eqfnfv2  5660  eqfnfvd  5662  eqfnfv2f  5663  fvreseq  5665  fnmptfvd  5666  fneqeql  5670  fconst2g  5777  cocan1  5834  cocan2  5835  tfri3  6425  updjud  7148  nninfwlporlemd  7238  ser0f  10626  prodf1f  11708  1arithlem4  12535  1arith  12536  isgrpinv  13186  cnmpt11  14519  cnmpt21  14527