Theorem funcoeqres 5614

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 5608	. . . 4 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺))
2	1	coeq2d 4892	. . 3 ⊢ (Fun 𝐺 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
3		coass 5255	. . . 4 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
4	3	eqcomi 2235	. . 3 ⊢ (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = ((𝐹 ∘ 𝐺) ∘ ◡𝐺)
5		coires1 5254	. . 3 ⊢ (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)
6	2, 4, 5	3eqtr3g 2287	. 2 ⊢ (Fun 𝐺 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺))
7		coeq1 4887	. 2 ⊢ ((𝐹 ∘ 𝐺) = 𝐻 → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐻 ∘ ◡𝐺))
8	6, 7	sylan9req 2285	1 ⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   I cid 4385  ^◡ccnv 4724  ran crn 4726   ↾ cres 4727   ∘ ccom 4729  Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-fun 5328

This theorem is referenced by: (None)