Theorem eqfunfv 5619

Description: Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Assertion

Ref	Expression
eqfunfv	|- ( ( Fun F /\ Fun G ) -> ( F = G <-> ( dom F = dom G /\ A. x e. dom F ( F ` x ) = ( G ` x ) ) ) )

Distinct variable groups:   x, F   x, G

Proof of Theorem eqfunfv

Step	Hyp	Ref	Expression
1		funfn 5247	. 2 |- ( Fun F <-> F Fn dom F )
2		funfn 5247	. 2 |- ( Fun G <-> G Fn dom G )
3		eqfnfv2 5615	. 2 |- ( ( F Fn dom F /\ G Fn dom G ) -> ( F = G <-> ( dom F = dom G /\ A. x e. dom F ( F ` x ) = ( G ` x ) ) ) )
4	1, 2, 3	syl2anb 291	1 |- ( ( Fun F /\ Fun G ) -> ( F = G <-> ( dom F = dom G /\ A. x e. dom F ( F ` x ) = ( G ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   A.wral 2455   dom cdm 4627   Fun wfun 5211   Fn wfn 5212   ` cfv 5217

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225

This theorem is referenced by:  ennnfonelemex  12415