Theorem funcoeqres 6863

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 6857	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺))
2	1	coeq2d 5861	. . 3 ⊢ (Fun 𝐺 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3		coass 6263	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
4	3	eqcomi 2739	. . 3 ⊢ (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = ((𝐹 ∘ 𝐺) ∘ ◡𝐺)
5		coires1 6262	. . 3 ⊢ (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
6	2, 4, 5	3eqtr3g 2793	. 2 ⊢ (Fun 𝐺 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺))
7		coeq1 5856	. 2 ⊢ ((𝐹 ∘ 𝐺) = 𝐻 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐻 ∘ ◡𝐺))
8	6, 7	sylan9req 2791	1 ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   I cid 5572  ^◡ccnv 5674  ran crn 5676   ↾ cres 5677   ∘ ccom 5679  Fun wfun 6536

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-fun 6544

This theorem is referenced by:  frlmup4  21575  evlseu  21865