Theorem eqfunfv 5758

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 5363	. 2 |- ( Fun F <-> F Fn dom F )
2		funfn 5363	. 2 |- ( Fun G <-> G Fn dom G )
3		eqfnfv2 5754	. 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 1398   A.wral 2511   dom cdm 4731   Fun wfun 5327   Fn wfn 5328   ` cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  ennnfonelemex  13115