Theorem funcoeqres 5645

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 5639	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺))
2	1	coeq2d 4917	. . 3 ⊢ (Fun 𝐺 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3		coass 5281	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
4	3	eqcomi 2236	. . 3 ⊢ (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = ((𝐹 ∘ 𝐺) ∘ ◡𝐺)
5		coires1 5280	. . 3 ⊢ (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
6	2, 4, 5	3eqtr3g 2288	. 2 ⊢ (Fun 𝐺 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺))
7		coeq1 4912	. 2 ⊢ ((𝐹 ∘ 𝐺) = 𝐻 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐻 ∘ ◡𝐺))
8	6, 7	sylan9req 2286	1 ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   I cid 4409  ^◡ccnv 4748  ran crn 4750   ↾ cres 4751   ∘ ccom 4753  Fun wfun 5346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-fun 5354

This theorem is referenced by: (None)