cocnv - Mathbox for Jeff Madsen

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Theorem cocnv 34994

Description: Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
cocnv	⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ↾ ran 𝐺))

**Proof of Theorem cocnv**
Step	Hyp	Ref	Expression
1		coass 6113	. 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ∘ (𝐺 ∘ ^◡𝐺))
2		funcocnv2 6634	. . . . 5 ⊢ (Fun 𝐺 → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
3	2	adantl 484	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
4	3	coeq2d 5728	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5		resco 6098	. . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6		funrel 6367	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
7		coi1 6110	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
8	6, 7	syl 17	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
9	8	reseq1d 5847	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
10	9	adantr 483	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
11	5, 10	syl5eqr 2870	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
12	4, 11	eqtrd 2856	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ↾ ran 𝐺))
13	1, 12	syl5eq 2868	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ↾ ran 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 I cid 5454 ^◡ccnv 5549 ran crn 5551 ↾ cres 5552 ∘ ccom 5554 Rel wrel 5555 Fun wfun 6344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-fun 6352

This theorem is referenced by: (None)