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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 5759 | . . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) | |
2 | cnvnonrel 39954 | . . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅ | |
3 | 2 | coeq1i 5733 | . . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴) |
4 | co01 6117 | . . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2851 | . . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
6 | 5 | cnveqi 5748 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅ |
7 | relco 6100 | . . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) | |
8 | dfrel2 6049 | . . 3 ⊢ (Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) ↔ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵))) | |
9 | 7, 8 | mpbi 232 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) |
10 | cnv0 6002 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2855 | 1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∖ cdif 3936 ∅c0 4294 ◡ccnv 5557 ∘ ccom 5562 Rel wrel 5563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 |
This theorem is referenced by: cnvtrcl0 39992 |
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