Theorem eqfunfv 6914

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 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺

Proof of Theorem eqfunfv

Step	Hyp	Ref	Expression
1		funfn 6464	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		funfn 6464	. 2 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3		eqfnfv2 6910	. 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐺 Fn dom 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥))))
4	1, 2, 3	syl2anb 598	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∀wral 3064  dom cdm 5589  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  fveqressseq  6957  fnprb  7084  fntpb  7085  symgfixf1  19045  nosepon  33868  nolesgn2ores  33875  nogesgn1ores  33877  nosupres  33910  nosupbnd2lem1  33918  noinfres  33925  noinfbnd2lem1  33933  noetasuplem4  33939  noetainflem4  33943  comptiunov2i  41314