Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 5756 | . . . 4 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) | |
2 | cnvnonrel 39968 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
3 | 2 | coeq2i 5731 | . . . 4 ⊢ (◡𝐵 ∘ ◡(𝐴 ∖ ◡◡𝐴)) = (◡𝐵 ∘ ∅) |
4 | co02 6113 | . . . 4 ⊢ (◡𝐵 ∘ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2848 | . . 3 ⊢ ◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
6 | 5 | cnveqi 5745 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ◡∅ |
7 | relco 6097 | . . 3 ⊢ Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) | |
8 | dfrel2 6046 | . . 3 ⊢ (Rel ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) ↔ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵)) | |
9 | 7, 8 | mpbi 232 | . 2 ⊢ ◡◡((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) |
10 | cnv0 5999 | . 2 ⊢ ◡∅ = ∅ | |
11 | 6, 9, 10 | 3eqtr3i 2852 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3933 ∅c0 4291 ◡ccnv 5554 ∘ ccom 5559 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 |
This theorem is referenced by: cnvtrcl0 40006 |
Copyright terms: Public domain | W3C validator |