cocnv - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cocnv		Structured version Visualization version GIF version

Theorem cocnv 32690

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 5571	. 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ∘ (𝐺 ∘ ^◡𝐺))
2		funcocnv2 6074	. . . . 5 ⊢ (Fun 𝐺 → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
3	2	adantl 481	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺 ∘ ^◡𝐺) = ( I ↾ ran 𝐺))
4	3	coeq2d 5206	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5		resco 5556	. . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6		funrel 5821	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
7		coi1 5568	. . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
8	6, 7	syl 17	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
9	8	reseq1d 5316	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
10	9	adantr 480	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
11	5, 10	syl5eqr 2658	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
12	4, 11	eqtrd 2644	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ^◡𝐺)) = (𝐹 ↾ ran 𝐺))
13	1, 12	syl5eq 2656	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ^◡𝐺) = (𝐹 ↾ ran 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 I cid 4948 ^◡ccnv 5037 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-fun 5806

This theorem is referenced by: (None)