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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 5454 | . . . 4 ⊢ ◡∅ = ∅ | |
2 | cnvco 5230 | . . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅) | |
3 | 1 | coeq2i 5204 | . . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅) |
4 | co02 5566 | . . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅ | |
5 | 2, 3, 4 | 3eqtri 2636 | . . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅ |
6 | 1, 5 | eqtr4i 2635 | . . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴) |
7 | 6 | cnveqi 5219 | . 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴) |
8 | rel0 5166 | . . 3 ⊢ Rel ∅ | |
9 | dfrel2 5502 | . . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅) | |
10 | 8, 9 | mpbi 219 | . 2 ⊢ ◡◡∅ = ∅ |
11 | relco 5550 | . . 3 ⊢ Rel (∅ ∘ 𝐴) | |
12 | dfrel2 5502 | . . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)) | |
13 | 11, 12 | mpbi 219 | . 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴) |
14 | 7, 10, 13 | 3eqtr3ri 2641 | 1 ⊢ (∅ ∘ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 |
This theorem is referenced by: xpcoid 5593 0trrel 13568 gsumval3 18131 utop2nei 21864 cononrel2 36920 |
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