![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > co01 | Structured version Visualization version GIF version |
Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
co01 | ⊢ (∅ ∘ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnv0 5844 | . . . 4 ⊢ ◡∅ = ∅ | |
2 | cnvco 5610 | . . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅) | |
3 | 1 | coeq2i 5585 | . . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅) |
4 | co02 5957 | . . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅ | |
5 | 2, 3, 4 | 3eqtri 2808 | . . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅ |
6 | 1, 5 | eqtr4i 2807 | . . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴) |
7 | 6 | cnveqi 5599 | . 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴) |
8 | rel0 5526 | . . 3 ⊢ Rel ∅ | |
9 | dfrel2 5891 | . . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅) | |
10 | 8, 9 | mpbi 222 | . 2 ⊢ ◡◡∅ = ∅ |
11 | relco 5941 | . . 3 ⊢ Rel (∅ ∘ 𝐴) | |
12 | dfrel2 5891 | . . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)) | |
13 | 11, 12 | mpbi 222 | . 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴) |
14 | 7, 10, 13 | 3eqtr3ri 2813 | 1 ⊢ (∅ ∘ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∅c0 4181 ◡ccnv 5410 ∘ ccom 5415 Rel wrel 5416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pr 5190 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-rab 3099 df-v 3419 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-op 4451 df-br 4935 df-opab 4997 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 |
This theorem is referenced by: xpcoid 5984 0trrel 14208 gsumval3 18793 utop2nei 22577 cononrel2 39358 |
Copyright terms: Public domain | W3C validator |