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Mirrors > Home > MPE Home > Th. List > cotrg | Structured version Visualization version GIF version |
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5666 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5666. (Revised by Richard Penner, 24-Dec-2019.) |
Ref | Expression |
---|---|
cotrg | ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 5275 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)} | |
2 | 1 | relopabi 5401 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) |
3 | ssrel 5364 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶)) |
5 | vex 3343 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 3343 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 5449 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) |
8 | df-br 4805 | . . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐶) | |
9 | 8 | bicomi 214 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝐶 ↔ 𝑥𝐶𝑧) |
10 | 7, 9 | imbi12i 339 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
11 | 19.23v 2020 | . . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
12 | 10, 11 | bitr4i 267 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
13 | 12 | albii 1896 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
14 | alcom 2186 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
15 | 13, 14 | bitri 264 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
16 | 15 | albii 1896 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
17 | 4, 16 | bitri 264 | 1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1630 ∃wex 1853 ∈ wcel 2139 ⊆ wss 3715 〈cop 4327 class class class wbr 4804 ∘ ccom 5270 Rel wrel 5271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-xp 5272 df-rel 5273 df-co 5275 |
This theorem is referenced by: cotr 5666 cotr2g 13936 |
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