Theorem eqfnfv 5777

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 5724	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		dffn5im 5724	. . 3 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	1, 2	eqeqan12d 2250	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) ) )
4		funfvex 5689	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
5	4	funfni 5460	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
6	5	ralrimiva 2617	. . . 4 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
7		mpteqb 5770	. . . 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 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   |-> cmpt 4173   Fn wfn 5349   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  eqfnfv2  5778  eqfnfvd  5780  eqfnfv2f  5781  fvreseq  5783  fnmptfvd  5784  fneqeql  5788  fconst2g  5901  cocan1  5962  cocan2  5963  tfri3  6600  updjud  7375  nninfwlporlemd  7465  ser0f  10900  prodf1f  12233  1arithlem4  13068  1arith  13069  isgrpinv  13784  psrbagconf1o  14845  cnmpt11  15165  cnmpt21  15173  nnnninfex  16817  nninfnfiinf  16818