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Mirrors > Home > MPE Home > Th. List > Mathboxes > cononrel2 | 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 |
---|---|
cononrel2 | ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5879 | . . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) | |
2 | cnvnonrel 42915 | . . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅ | |
3 | 2 | coeq1i 5853 | . . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴) |
4 | co01 6254 | . . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2758 | . . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
6 | 5 | cnveqi 5868 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅ |
7 | relco 6101 | . . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) | |
8 | dfrel2 6182 | . . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) |
10 | cnv0 6134 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2762 | 1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3940 ∅c0 4317 ◡ccnv 5668 ∘ ccom 5673 Rel wrel 5674 |
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-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 |
This theorem is referenced by: cnvtrcl0 42953 |
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