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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 5549 | . . . . . 6 ⊢ dom 𝐴 = ran ◡𝐴 | |
2 | 1 | sseq1i 3855 | . . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ◡𝐴 ⊆ 𝐶) |
3 | cores 5880 | . . . . 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 5541 | . . . . 5 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
6 | cocnvcnv1 5888 | . . . . 5 ⊢ (◡◡(◡𝐵 ↾ 𝐶) ∘ ◡𝐴) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) | |
7 | 5, 6 | eqtri 2850 | . . . 4 ⊢ ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ((◡𝐵 ↾ 𝐶) ∘ ◡𝐴) |
8 | cnvco 5541 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
9 | 4, 7, 8 | 3eqtr4g 2887 | . . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡(𝐴 ∘ 𝐵)) |
10 | 9 | cnveqd 5531 | . 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ◡◡(𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = ◡◡(𝐴 ∘ 𝐵)) |
11 | relco 5875 | . . 3 ⊢ Rel (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) | |
12 | dfrel2 5825 | . . 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 5875 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
15 | dfrel2 5825 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)) | |
16 | 14, 15 | mpbi 222 | . 2 ⊢ ◡◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵) |
17 | 10, 13, 16 | 3eqtr3g 2885 | 1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ⊆ wss 3799 ◡ccnv 5342 dom cdm 5343 ran crn 5344 ↾ cres 5345 ∘ ccom 5347 Rel wrel 5348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 |
This theorem is referenced by: fcoi1 6316 ofco2 20626 |
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