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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 4983 | . . . 4 ⊢ (𝑆 ↾ dom 𝑅) ⊆ 𝑆 | |
| 2 | dmss 4877 | . . . 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 5195 | . . 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 5145 | . . . 4 ⊢ (◡◡𝑅 ↾ dom 𝑆) = (𝑅 ↾ dom 𝑆) | |
| 7 | 6 | cnveqi 4853 | . . 3 ⊢ ◡(◡◡𝑅 ↾ dom 𝑆) = ◡(𝑅 ↾ dom 𝑆) |
| 8 | 7 | coeq2i 4838 | . 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
| 9 | dfdm4 4870 | . . . 4 ⊢ dom 𝑅 = ran ◡𝑅 | |
| 10 | 9 | eqimss2i 3250 | . . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅 |
| 11 | cores 5186 | . . 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 2235 | 1 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆)) |
| Colors of variables: wff set class |
| Syntax hints: = wceq 1373 ⊆ wss 3166 ◡ccnv 4674 dom cdm 4675 ran crn 4676 ↾ cres 4677 ∘ ccom 4679 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-br 4045 df-opab 4106 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 |
| This theorem is referenced by: cocnvss 5208 |
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