Theorem eqfnfv 5800

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 5745	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		dffn5im 5745	. . 3 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	1, 2	eqeqan12d 2254	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( G ` x ) ) ) )
4		funfvex 5710	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
5	4	funfni 5481	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
6	5	ralrimiva 2623	. . . 4 |- ( F Fn A -> A. x e. A ( F ` x ) e. _V )
7		mpteqb 5793	. . . 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 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   |-> cmpt 4190   Fn wfn 5370   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  eqfnfv2  5801  eqfnfvd  5803  eqfnfv2f  5804  fvreseq  5806  fnmptfvd  5807  fneqeql  5811  fconst2g  5924  cocan1  5986  cocan2  5987  tfri3  6631  updjud  7415  nninfwlporlemd  7505  ser0f  10952  hashf1lem1  11266  prodf1f  12291  1arithlem4  13126  1arith  13127  isgrpinv  13839  psrbagconf1o  14990  cnmpt11  15310  cnmpt21  15318  nnnninfex  16973  nninfnfiinf  16974