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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 6111 | . 2 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺)) | |
2 | funcocnv2 6632 | . . . . 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 5726 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺))) |
5 | resco 6096 | . . . 4 ⊢ ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺)) | |
6 | funrel 6365 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
7 | coi1 6108 | . . . . . . 7 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
9 | 8 | reseq1d 5845 | . . . . 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 2867 | . . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺)) |
12 | 4, 11 | eqtrd 2853 | . 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ↾ ran 𝐺)) |
13 | 1, 12 | syl5eq 2865 | 1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 I cid 5452 ◡ccnv 5547 ran crn 5549 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-fun 6350 |
This theorem is referenced by: (None) |
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