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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 5966 | . . . 4 ⊢ ◡∅ = ∅ | |
2 | cnvco 5720 | . . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅) | |
3 | 1 | coeq2i 5695 | . . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅) |
4 | co02 6080 | . . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅ | |
5 | 2, 3, 4 | 3eqtri 2825 | . . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅ |
6 | 1, 5 | eqtr4i 2824 | . . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴) |
7 | 6 | cnveqi 5709 | . 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴) |
8 | rel0 5636 | . . 3 ⊢ Rel ∅ | |
9 | dfrel2 6013 | . . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅) | |
10 | 8, 9 | mpbi 233 | . 2 ⊢ ◡◡∅ = ∅ |
11 | relco 6064 | . . 3 ⊢ Rel (∅ ∘ 𝐴) | |
12 | dfrel2 6013 | . . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)) | |
13 | 11, 12 | mpbi 233 | . 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴) |
14 | 7, 10, 13 | 3eqtr3ri 2830 | 1 ⊢ (∅ ∘ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 ◡ccnv 5518 ∘ ccom 5523 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 |
This theorem is referenced by: xpcoid 6109 0trrel 14332 relexpsucrd 14384 relexpaddd 14405 gsumval3 19020 utop2nei 22856 cononrel2 40295 |
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