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Mirrors > Home > ILE Home > Th. List > cores2 | GIF version |
Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Ref | Expression |
---|---|
cores2 | ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 4731 | . . . . . 6 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | 1 | sseq1i 3123 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ◡𝐴 ⊆ 𝐶) |
3 | cores 5042 | . . . . 5 ⊢ (ran ◡𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) | |
4 | 2, 3 | sylbi 120 | . . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) |
5 | cnvco 4724 | . . . . 5 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
6 | cocnvcnv1 5049 | . . . . 5 ⊢ (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
7 | 5, 6 | eqtri 2160 | . . . 4 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) |
8 | cnvco 4724 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 4, 7, 8 | 3eqtr4g 2197 | . . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡(𝐴 ∘ 𝐵)) |
10 | 9 | cnveqd 4715 | . 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡◡(𝐴 ∘ 𝐵)) |
11 | relco 5037 | . . 3 ⊢ Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) | |
12 | dfrel2 4989 | . . 3 ⊢ (Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) ↔ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶))) | |
13 | 11, 12 | mpbi 144 | . 2 ⊢ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) |
14 | relco 5037 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
15 | dfrel2 4989 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
16 | 14, 15 | mpbi 144 | . 2 ⊢ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
17 | 10, 13, 16 | 3eqtr3g 2195 | 1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⊆ wss 3071 ◡ccnv 4538 dom cdm 4539 ran crn 4540 ↾ cres 4541 ∘ ccom 4543 Rel wrel 4544 |
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: cocnvres 5063 fcoi1 5303 |
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