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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 5902 | . . . . . 6 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | 1 | sseq1i 4010 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ◡𝐴 ⊆ 𝐶) |
3 | cores 6258 | . . . . 5 ⊢ (ran ◡𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) | |
4 | 2, 3 | sylbi 216 | . . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) |
5 | cnvco 5892 | . . . . 5 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
6 | cocnvcnv1 6266 | . . . . 5 ⊢ (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
7 | 5, 6 | eqtri 2756 | . . . 4 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) |
8 | cnvco 5892 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 4, 7, 8 | 3eqtr4g 2793 | . . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡(𝐴 ∘ 𝐵)) |
10 | 9 | cnveqd 5882 | . 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡◡(𝐴 ∘ 𝐵)) |
11 | relco 6117 | . . 3 ⊢ Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) | |
12 | dfrel2 6198 | . . 3 ⊢ (Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) ↔ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶))) | |
13 | 11, 12 | mpbi 229 | . 2 ⊢ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) |
14 | relco 6117 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
15 | dfrel2 6198 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
16 | 14, 15 | mpbi 229 | . 2 ⊢ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
17 | 10, 13, 16 | 3eqtr3g 2791 | 1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⊆ wss 3949 ◡ccnv 5681 dom cdm 5682 ran crn 5683 ↾ cres 5684 ∘ ccom 5686 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 |
This theorem is referenced by: fcoi1 6776 ofco2 22373 cycpmconjvlem 32883 |
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