Theorem eqfnfv 5732

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 5679	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		dffn5im 5679	. . 3 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	1, 2	eqeqan12d 2245	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) ) )
4		funfvex 5644	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
5	4	funfni 5423	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
6	5	ralrimiva 2603	. . . 4 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
7		mpteqb 5725	. . . 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 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  eqfnfv2  5733  eqfnfvd  5735  eqfnfv2f  5736  fvreseq  5738  fnmptfvd  5739  fneqeql  5743  fconst2g  5854  cocan1  5911  cocan2  5912  tfri3  6513  updjud  7249  nninfwlporlemd  7339  ser0f  10756  prodf1f  12054  1arithlem4  12889  1arith  12890  isgrpinv  13587  cnmpt11  14957  cnmpt21  14965  nnnninfex  16388  nninfnfiinf  16389