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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 5939 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5939. (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 5528 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)} | |
2 | 1 | relopabi 5658 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) |
3 | ssrel 5621 | . . 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 3444 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 3444 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 5706 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) |
8 | df-br 5031 | . . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐶) | |
9 | 8 | bicomi 227 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝐶 ↔ 𝑥𝐶𝑧) |
10 | 7, 9 | imbi12i 354 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
11 | 19.23v 1943 | . . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
12 | 10, 11 | bitr4i 281 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
13 | 12 | albii 1821 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
14 | alcom 2160 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
15 | 13, 14 | bitri 278 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
16 | 15 | albii 1821 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
17 | 4, 16 | bitri 278 | 1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 class class class wbr 5030 ∘ 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-co 5528 |
This theorem is referenced by: cotr 5939 cotr2g 14327 |
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