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Mirrors > Home > MPE Home > Th. List > Mathboxes > cononrel1 | 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 |
---|---|
cononrel1 | ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5308 | . . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) | |
2 | cnvnonrel 37894 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 2 | coeq2i 5282 | . . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅) |
4 | co02 5649 | . . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2648 | . . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
6 | 5 | cnveqi 5297 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅ |
7 | relco 5633 | . . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) | |
8 | dfrel2 5583 | . . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)) | |
9 | 7, 8 | mpbi 220 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) |
10 | cnv0 5535 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2652 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∖ cdif 3571 ∅c0 3915 ◡ccnv 5113 ∘ ccom 5118 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 |
This theorem is referenced by: cnvtrcl0 37933 |
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