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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 5739 | . . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) | |
2 | cnvnonrel 40813 | . . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅ | |
3 | 2 | coeq1i 5713 | . . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴) |
4 | co01 6105 | . . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2763 | . . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
6 | 5 | cnveqi 5728 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅ |
7 | relco 6088 | . . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) | |
8 | dfrel2 6032 | . . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))) | |
9 | 7, 8 | mpbi 233 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) |
10 | cnv0 5984 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2767 | 1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∖ cdif 3850 ∅c0 4223 ◡ccnv 5535 ∘ ccom 5540 Rel wrel 5541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 |
This theorem is referenced by: cnvtrcl0 40851 |
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