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Theorem funcoeqres 5157
 Description: Re-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 5151 . . . 4 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
21coeq2d 4498 . . 3 (Fun 𝐺 → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3 coass 4839 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
43eqcomi 2044 . . 3 (𝐹 ∘ (𝐺𝐺)) = ((𝐹𝐺) ∘ 𝐺)
5 coires1 4838 . . 3 (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
62, 4, 53eqtr3g 2095 . 2 (Fun 𝐺 → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
7 coeq1 4493 . 2 ((𝐹𝐺) = 𝐻 → ((𝐹𝐺) ∘ 𝐺) = (𝐻𝐺))
86, 7sylan9req 2093 1 ((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   I cid 4025  ◡ccnv 4344  ran crn 4346   ↾ cres 4347   ∘ ccom 4349  Fun wfun 4896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904 This theorem is referenced by: (None)
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