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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 5755 | . . . 4 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) | |
2 | cnvnonrel 39946 | . . . . 5 ⊢ ◡(𝐵 ∖ ◡◡𝐵) = ∅ | |
3 | 2 | coeq1i 5729 | . . . 4 ⊢ (◡(𝐵 ∖ ◡◡𝐵) ∘ ◡𝐴) = (∅ ∘ ◡𝐴) |
4 | co01 6113 | . . . 4 ⊢ (∅ ∘ ◡𝐴) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2848 | . . 3 ⊢ ◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
6 | 5 | cnveqi 5744 | . 2 ⊢ ◡◡(𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ◡∅ |
7 | relco 6096 | . . 3 ⊢ Rel (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) | |
8 | dfrel2 6045 | . . 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 5998 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2852 | 1 ⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3932 ∅c0 4290 ◡ccnv 5553 ∘ ccom 5558 Rel wrel 5559 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 |
This theorem is referenced by: cnvtrcl0 39984 |
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