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Theorem funcoeqres 6648
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 6642 . . . 4 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
21coeq2d 5736 . . 3 (Fun 𝐺 → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3 coass 6121 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
43eqcomi 2833 . . 3 (𝐹 ∘ (𝐺𝐺)) = ((𝐹𝐺) ∘ 𝐺)
5 coires1 6120 . . 3 (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
62, 4, 53eqtr3g 2882 . 2 (Fun 𝐺 → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
7 coeq1 5731 . 2 ((𝐹𝐺) = 𝐻 → ((𝐹𝐺) ∘ 𝐺) = (𝐻𝐺))
86, 7sylan9req 2880 1 ((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536   I cid 5462  ccnv 5557  ran crn 5559  cres 5560  ccom 5562  Fun wfun 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-fun 6360
This theorem is referenced by:  evlseu  20299  frlmup4  20948
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