Theorem eqfunfv 5634

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 5261	. 2 |- ( Fun F <-> F Fn dom F )
2		funfn 5261	. 2 |- ( Fun G <-> G Fn dom G )
3		eqfnfv2 5630	. 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 1364   A.wral 2468   dom cdm 4641   Fun wfun 5225   Fn wfn 5226   ` cfv 5231

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239

This theorem is referenced by:  ennnfonelemex  12433