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Theorem funcoeqres 6865
Description: Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
funcoeqres ((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))

Proof of Theorem funcoeqres
StepHypRef Expression
1 funcocnv2 6859 . . . 4 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
21coeq2d 5863 . . 3 (Fun 𝐺 → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3 coass 6265 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
43eqcomi 2742 . . 3 (𝐹 ∘ (𝐺𝐺)) = ((𝐹𝐺) ∘ 𝐺)
5 coires1 6264 . . 3 (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
62, 4, 53eqtr3g 2796 . 2 (Fun 𝐺 → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
7 coeq1 5858 . 2 ((𝐹𝐺) = 𝐻 → ((𝐹𝐺) ∘ 𝐺) = (𝐻𝐺))
86, 7sylan9req 2794 1 ((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542   I cid 5574  ccnv 5676  ran crn 5678  cres 5679  ccom 5681  Fun wfun 6538
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-fun 6546
This theorem is referenced by:  frlmup4  21356  evlseu  21646
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