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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 6160 | . . . 4 ⊢ ◡∅ = ∅ | |
| 2 | cnvco 5896 | . . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅) | |
| 3 | 1 | coeq2i 5871 | . . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅) |
| 4 | co02 6280 | . . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅ | |
| 5 | 2, 3, 4 | 3eqtri 2769 | . . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅ |
| 6 | 1, 5 | eqtr4i 2768 | . . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴) |
| 7 | 6 | cnveqi 5885 | . 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴) |
| 8 | rel0 5809 | . . 3 ⊢ Rel ∅ | |
| 9 | dfrel2 6209 | . . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅) | |
| 10 | 8, 9 | mpbi 230 | . 2 ⊢ ◡◡∅ = ∅ |
| 11 | relco 6126 | . . 3 ⊢ Rel (∅ ∘ 𝐴) | |
| 12 | dfrel2 6209 | . . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)) | |
| 13 | 11, 12 | mpbi 230 | . 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴) |
| 14 | 7, 10, 13 | 3eqtr3ri 2774 | 1 ⊢ (∅ ∘ 𝐴) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∅c0 4333 ◡ccnv 5684 ∘ ccom 5689 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 |
| This theorem is referenced by: xpcoid 6310 0trrel 15020 relexpsucrd 15072 relexpaddd 15093 gsumval3 19925 utop2nei 24259 cononrel2 43608 |
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