cocnv - Mathbox for Jeff Madsen

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Theorem cocnv 33831

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 5813	. 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ∘ (𝐺 ∘ ^◡𝐺))
2		funcocnv2 6320	. . . . 5 ⊢ (Fun 𝐺 → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
3	2	adantl 473	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
4	3	coeq2d 5438	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5		resco 5798	. . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6		funrel 6064	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
7		coi1 5810	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
8	6, 7	syl 17	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
9	8	reseq1d 5548	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
10	9	adantr 472	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
11	5, 10	syl5eqr 2806	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
12	4, 11	eqtrd 2792	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ↾ ran 𝐺))
13	1, 12	syl5eq 2804	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ↾ ran 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1630 I cid 5171 ^◡ccnv 5263 ran crn 5265 ↾ cres 5266 ∘ ccom 5268 Rel wrel 5269 Fun wfun 6041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pr 5053

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ral 3053 df-rex 3054 df-rab 3057 df-v 3340 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-sn 4320 df-pr 4322 df-op 4326 df-br 4803 df-opab 4863 df-id 5172 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-fun 6049

This theorem is referenced by: (None)