cocnv - Mathbox for Jeff Madsen

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Theorem cocnv 33520

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 5654	. 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ∘ (𝐺 ∘ ^◡𝐺))
2		funcocnv2 6161	. . . . 5 ⊢ (Fun 𝐺 → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
3	2	adantl 482	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
4	3	coeq2d 5284	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5		resco 5639	. . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6		funrel 5905	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
7		coi1 5651	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
8	6, 7	syl 17	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
9	8	reseq1d 5395	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
10	9	adantr 481	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
11	5, 10	syl5eqr 2670	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
12	4, 11	eqtrd 2656	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ↾ ran 𝐺))
13	1, 12	syl5eq 2668	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ↾ ran 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 I cid 5023 ^◡ccnv 5113 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 Rel wrel 5119 Fun wfun 5882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-fun 5890

This theorem is referenced by: (None)