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Mirrors > Home > MPE Home > Th. List > cores2 | Structured version Visualization version 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 5749 | . . . . . 6 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | 1 | sseq1i 3915 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ◡𝐴 ⊆ 𝐶) |
3 | cores 6093 | . . . . 5 ⊢ (ran ◡𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) | |
4 | 2, 3 | sylbi 220 | . . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) |
5 | cnvco 5739 | . . . . 5 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
6 | cocnvcnv1 6101 | . . . . 5 ⊢ (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
7 | 5, 6 | eqtri 2759 | . . . 4 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) |
8 | cnvco 5739 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 4, 7, 8 | 3eqtr4g 2796 | . . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡(𝐴 ∘ 𝐵)) |
10 | 9 | cnveqd 5729 | . 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡◡(𝐴 ∘ 𝐵)) |
11 | relco 6088 | . . 3 ⊢ Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) | |
12 | dfrel2 6032 | . . 3 ⊢ (Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) ↔ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶))) | |
13 | 11, 12 | mpbi 233 | . 2 ⊢ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) |
14 | relco 6088 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
15 | dfrel2 6032 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
16 | 14, 15 | mpbi 233 | . 2 ⊢ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
17 | 10, 13, 16 | 3eqtr3g 2794 | 1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 ↾ cres 5538 ∘ ccom 5540 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 |
This theorem is referenced by: fcoi1 6571 ofco2 21302 cycpmconjvlem 31081 |
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