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Theorem eqfunfv 7055
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(𝐹𝑥) = (𝐺𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺

Proof of Theorem eqfunfv
StepHypRef Expression
1 funfn 6597 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 funfn 6597 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
3 eqfnfv2 7051 . 2 ((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
41, 2, 3syl2anb 598 1 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wral 3058  dom cdm 5688  Fun wfun 6556   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  fveqressseq  7098  fnprb  7227  fntpb  7228  symgfixf1  19469  nosepon  27724  nolesgn2ores  27731  nogesgn1ores  27733  nosupres  27766  nosupbnd2lem1  27774  noinfres  27781  noinfbnd2lem1  27789  noetasuplem4  27795  noetainflem4  27799  comptiunov2i  43695
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