![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6077 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
2 | relcnv 5934 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
3 | coi1 6082 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
5 | 1, 4 | eqtr3i 2823 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
6 | 5 | reseq1i 5814 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
7 | resco 6070 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
8 | 6, 7 | eqtr3i 2823 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
9 | rescnvcnv 6028 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
10 | 8, 9 | eqtr3i 2823 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 I cid 5424 ◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 |
This theorem is referenced by: relcoi1 6097 funcoeqres 6620 relexpaddg 14404 lindfres 20512 lindsmm 20517 psrass1lem 20615 kgencn2 22162 ustssco 22820 symgcom 30777 cycpmconjv 30834 cycpmconjslem1 30846 erdsze2lem2 32564 poimirlem9 35066 mzpresrename 39691 diophrw 39700 eldioph2 39703 diophren 39754 relexpiidm 40405 relexpaddss 40419 cotrclrcl 40443 funcrngcsetcALT 44623 itcoval1 45077 |
Copyright terms: Public domain | W3C validator |