Theorem funcoeqres 5473

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 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))

Proof of Theorem funcoeqres

Step	Hyp	Ref	Expression
1		funcocnv2 5467	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺))
2	1	coeq2d 4773	. . 3 ⊢ (Fun 𝐺 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3		coass 5129	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
4	3	eqcomi 2174	. . 3 ⊢ (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = ((𝐹 ∘ 𝐺) ∘ ◡𝐺)
5		coires1 5128	. . 3 ⊢ (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
6	2, 4, 5	3eqtr3g 2226	. 2 ⊢ (Fun 𝐺 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺))
7		coeq1 4768	. 2 ⊢ ((𝐹 ∘ 𝐺) = 𝐻 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐻 ∘ ◡𝐺))
8	6, 7	sylan9req 2224	1 ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   I cid 4273  ^◡ccnv 4610  ran crn 4612   ↾ cres 4613   ∘ ccom 4615  Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-fun 5200

This theorem is referenced by: (None)