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| Mirrors > Home > MPE Home > Th. List > coires1 | Structured version Visualization version GIF version | ||
| Description: Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
| Ref | Expression |
|---|---|
| coires1 | ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cocnvcnv1 6277 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
| 2 | relcnv 6122 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
| 3 | coi1 6282 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
| 4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
| 5 | 1, 4 | eqtr3i 2767 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
| 6 | 5 | reseq1i 5993 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
| 7 | resco 6270 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
| 8 | 6, 7 | eqtr3i 2767 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
| 9 | rescnvcnv 6224 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
| 10 | 8, 9 | eqtr3i 2767 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 I cid 5577 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 |
| This theorem is referenced by: relcoi1 6298 funcoeqres 6879 f1ofvswap 7326 relexpaddg 15092 funcrngcsetcALT 20641 lindfres 21843 lindsmm 21848 psrass1lem 21952 kgencn2 23565 ustssco 24223 symgcom 33103 cycpmconjv 33162 cycpmconjslem1 33174 erdsze2lem2 35209 poimirlem9 37636 mzpresrename 42761 diophrw 42770 eldioph2 42773 diophren 42824 relexpiidm 43717 relexpaddss 43731 cotrclrcl 43755 itcoval1 48584 |
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