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Theorem funcoeqres 5268
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 5262 . . . 4 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
21coeq2d 4586 . . 3 (Fun 𝐺 → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3 coass 4936 . . . 4 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
43eqcomi 2092 . . 3 (𝐹 ∘ (𝐺𝐺)) = ((𝐹𝐺) ∘ 𝐺)
5 coires1 4935 . . 3 (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
62, 4, 53eqtr3g 2143 . 2 (Fun 𝐺 → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
7 coeq1 4581 . 2 ((𝐹𝐺) = 𝐻 → ((𝐹𝐺) ∘ 𝐺) = (𝐻𝐺))
86, 7sylan9req 2141 1 ((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289   I cid 4106  ccnv 4427  ran crn 4429  cres 4430  ccom 4432  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-fun 5004
This theorem is referenced by: (None)
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