![]() |
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 6257 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
2 | relcnv 6104 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
3 | coi1 6262 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
5 | 1, 4 | eqtr3i 2763 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
6 | 5 | reseq1i 5978 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
7 | resco 6250 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
8 | 6, 7 | eqtr3i 2763 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
9 | rescnvcnv 6204 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
10 | 8, 9 | eqtr3i 2763 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 I cid 5574 ◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 |
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-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 |
This theorem is referenced by: relcoi1 6278 funcoeqres 6865 f1ofvswap 7304 relexpaddg 15000 lindfres 21378 lindsmm 21383 psrass1lemOLD 21493 psrass1lem 21496 kgencn2 23061 ustssco 23719 symgcom 32244 cycpmconjv 32301 cycpmconjslem1 32313 erdsze2lem2 34195 poimirlem9 36497 mzpresrename 41488 diophrw 41497 eldioph2 41500 diophren 41551 relexpiidm 42455 relexpaddss 42469 cotrclrcl 42493 funcrngcsetcALT 46897 itcoval1 47349 |
Copyright terms: Public domain | W3C validator |