Theorem eqfunfv 7066

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 6598	. 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		funfn 6598	. 2 ⊢ (Fun 𝐺 ↔ 𝐺 Fn dom 𝐺)
3		eqfnfv2 7061	. 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 610	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3080  dom cdm 5658  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574

This theorem is used by:  fveqressseq  7111  fnprb  7245  fntpb  7246  symgfixf1  19702  nosepon  28079  nolesgn2ores  28086  nogesgn1ores  28088  nosupres  28121  nosupbnd2lem1  28129  noinfres  28136  noinfbnd2lem1  28144  noetasuplem4  28150  noetainflem4  28154  comptiunov2i  44967