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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 4851 | . . . 4 ⊢ (𝑆 ↾ dom 𝑅) ⊆ 𝑆 | |
2 | dmss 4746 | . . . 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 5059 | . . 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 5009 | . . . 4 ⊢ (◡◡𝑅 ↾ dom 𝑆) = (𝑅 ↾ dom 𝑆) | |
7 | 6 | cnveqi 4722 | . . 3 ⊢ ◡(◡◡𝑅 ↾ dom 𝑆) = ◡(𝑅 ↾ dom 𝑆) |
8 | 7 | coeq2i 4707 | . 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
9 | dfdm4 4739 | . . . 4 ⊢ dom 𝑅 = ran ◡𝑅 | |
10 | 9 | eqimss2i 3159 | . . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅 |
11 | cores 5050 | . . 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 2170 | 1 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ⊆ wss 3076 ◡ccnv 4546 dom cdm 4547 ran crn 4548 ↾ cres 4549 ∘ ccom 4551 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 |
This theorem is referenced by: cocnvss 5072 |
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