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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 5899 | . . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) | |
2 | cnvnonrel 43578 | . . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅ | |
3 | 2 | coeq1i 5873 | . . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴) |
4 | co01 6283 | . . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2767 | . . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
6 | 5 | cnveqi 5888 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅ |
7 | relco 6129 | . . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) | |
8 | dfrel2 6211 | . . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))) | |
9 | 7, 8 | mpbi 230 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) |
10 | cnv0 6163 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2771 | 1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∅c0 4339 ◡ccnv 5688 ∘ ccom 5693 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 |
This theorem is referenced by: cnvtrcl0 43616 |
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