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Mirrors > Home > ILE Home > Th. List > co01 | GIF version |
Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
co01 | ⊢ (∅ ∘ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnv0 5012 | . . . 4 ⊢ ◡∅ = ∅ | |
2 | cnvco 4794 | . . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅) | |
3 | 1 | coeq2i 4769 | . . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅) |
4 | co02 5122 | . . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅ | |
5 | 2, 3, 4 | 3eqtri 2195 | . . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅ |
6 | 1, 5 | eqtr4i 2194 | . . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴) |
7 | 6 | cnveqi 4784 | . 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴) |
8 | rel0 4734 | . . 3 ⊢ Rel ∅ | |
9 | dfrel2 5059 | . . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅) | |
10 | 8, 9 | mpbi 144 | . 2 ⊢ ◡◡∅ = ∅ |
11 | relco 5107 | . . 3 ⊢ Rel (∅ ∘ 𝐴) | |
12 | dfrel2 5059 | . . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)) | |
13 | 11, 12 | mpbi 144 | . 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴) |
14 | 7, 10, 13 | 3eqtr3ri 2200 | 1 ⊢ (∅ ∘ 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∅c0 3414 ◡ccnv 4608 ∘ ccom 4613 Rel wrel 4614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 |
This theorem is referenced by: (None) |
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