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Mirrors > Home > ILE Home > Th. List > cocnvres | GIF version |
Description: Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
cocnvres | ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resss 4843 | . . . 4 ⊢ (𝑆 ↾ dom 𝑅) ⊆ 𝑆 | |
2 | dmss 4738 | . . . 4 ⊢ ((𝑆 ↾ dom 𝑅) ⊆ 𝑆 → dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆 |
4 | cores2 5051 | . . 3 ⊢ (dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆 → ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅) |
6 | rescnvcnv 5001 | . . . 4 ⊢ (◡◡𝑅 ↾ dom 𝑆) = (𝑅 ↾ dom 𝑆) | |
7 | 6 | cnveqi 4714 | . . 3 ⊢ ◡(◡◡𝑅 ↾ dom 𝑆) = ◡(𝑅 ↾ dom 𝑆) |
8 | 7 | coeq2i 4699 | . 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
9 | dfdm4 4731 | . . . 4 ⊢ dom 𝑅 = ran ◡𝑅 | |
10 | 9 | eqimss2i 3154 | . . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅 |
11 | cores 5042 | . . 3 ⊢ (ran ◡𝑅 ⊆ dom 𝑅 → ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅) = (𝑆 ∘ ◡𝑅)) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅) = (𝑆 ∘ ◡𝑅) |
13 | 5, 8, 12 | 3eqtr3ri 2169 | 1 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3071 ◡ccnv 4538 dom cdm 4539 ran crn 4540 ↾ cres 4541 ∘ ccom 4543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 |
This theorem is referenced by: cocnvss 5064 |
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