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Mirrors > Home > MPE Home > Th. List > Mathboxes > cocnv | Structured version Visualization version GIF version |
Description: Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
cocnv | ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coass 6221 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺)) | |
2 | funcocnv2 6813 | . . . . 5 ⊢ (Fun 𝐺 → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺)) | |
3 | 2 | adantl 483 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺 ∘ ◡𝐺) = ( I ↾ ran 𝐺)) |
4 | 3 | coeq2d 5822 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺))) |
5 | resco 6206 | . . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺)) | |
6 | funrel 6522 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
7 | coi1 6218 | . . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
9 | 8 | reseq1d 5940 | . . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺)) |
10 | 9 | adantr 482 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺)) |
11 | 5, 10 | eqtr3id 2787 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)) |
12 | 4, 11 | eqtrd 2773 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ↾ ran 𝐺)) |
13 | 1, 12 | eqtrid 2785 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 I cid 5534 ◡ccnv 5636 ran crn 5638 ↾ cres 5639 ∘ ccom 5641 Rel wrel 5642 Fun wfun 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-fun 6502 |
This theorem is referenced by: (None) |
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