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Mirrors > Home > MPE Home > Th. List > Mathboxes > cononrel1 | Structured version Visualization version GIF version |
Description: Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
Ref | Expression |
---|---|
cononrel1 | ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5888 | . . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) | |
2 | cnvnonrel 43018 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 2 | coeq2i 5863 | . . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅) |
4 | co02 6264 | . . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2760 | . . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
6 | 5 | cnveqi 5877 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅ |
7 | relco 6112 | . . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) | |
8 | dfrel2 6193 | . . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) |
10 | cnv0 6145 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2764 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∖ cdif 3944 ∅c0 4323 ◡ccnv 5677 ∘ ccom 5682 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 |
This theorem is referenced by: cnvtrcl0 43056 |
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