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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 5517 | . . . . . 6 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | 1 | sseq1i 3823 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ◡𝐴 ⊆ 𝐶) |
3 | cores 5855 | . . . . 5 ⊢ (ran ◡𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) | |
4 | 2, 3 | sylbi 209 | . . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = (◡𝐵 ∘ ◡𝐴)) |
5 | cnvco 5509 | . . . . 5 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
6 | cocnvcnv1 5863 | . . . . 5 ⊢ (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
7 | 5, 6 | eqtri 2819 | . . . 4 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) |
8 | cnvco 5509 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 4, 7, 8 | 3eqtr4g 2856 | . . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡(𝐴 ∘ 𝐵)) |
10 | 9 | cnveqd 5499 | . 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡◡(𝐴 ∘ 𝐵)) |
11 | relco 5850 | . . 3 ⊢ Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) | |
12 | dfrel2 5798 | . . 3 ⊢ (Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) ↔ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶))) | |
13 | 11, 12 | mpbi 222 | . 2 ⊢ ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) |
14 | relco 5850 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
15 | dfrel2 5798 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
16 | 14, 15 | mpbi 222 | . 2 ⊢ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
17 | 10, 13, 16 | 3eqtr3g 2854 | 1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ⊆ wss 3767 ◡ccnv 5309 dom cdm 5310 ran crn 5311 ↾ cres 5312 ∘ ccom 5314 Rel wrel 5315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 |
This theorem is referenced by: fcoi1 6291 ofco2 20580 |
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