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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 6112 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
2 | relcnv 5969 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
3 | coi1 6117 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
5 | 1, 4 | eqtr3i 2848 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
6 | 5 | reseq1i 5851 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
7 | resco 6105 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
8 | 6, 7 | eqtr3i 2848 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
9 | rescnvcnv 6063 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
10 | 8, 9 | eqtr3i 2848 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 I cid 5461 ◡ccnv 5556 ↾ cres 5559 ∘ ccom 5561 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 |
This theorem is referenced by: relcoi1 6131 funcoeqres 6647 relexpaddg 14414 psrass1lem 20159 lindfres 20969 lindsmm 20974 kgencn2 22167 ustssco 22825 symgcom 30729 cycpmconjv 30786 cycpmconjslem1 30798 erdsze2lem2 32453 poimirlem9 34903 mzpresrename 39354 diophrw 39363 eldioph2 39366 diophren 39417 relexpiidm 40056 relexpaddss 40070 cotrclrcl 40094 funcrngcsetcALT 44277 |
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