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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 5427 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5427. (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 5047 | . . . 4 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)} | |
2 | 1 | relopabi 5167 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) |
3 | ssrel 5130 | . . 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 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 3176 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 5215 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) |
8 | df-br 4584 | . . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐶) | |
9 | 8 | bicomi 213 | . . . . . . 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 1889 | . . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
12 | 10, 11 | bitr4i 266 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
13 | 12 | albii 1737 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
14 | alcom 2024 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
15 | 13, 14 | bitri 263 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
16 | 15 | albii 1737 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
17 | 4, 16 | bitri 263 | 1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 ∘ ccom 5042 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-co 5047 |
This theorem is referenced by: cotr 5427 cotr2g 13563 |
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