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| Mirrors > Home > MPE Home > Th. List > cotrgOLDOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of cotrg 6127 as of 19-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6130. (Revised by Richard Penner, 24-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cotrgOLDOLD | ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relco 6126 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
| 2 | ssrel 5792 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶))) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶)) | 
| 4 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 5 | vex 3484 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
| 6 | 4, 5 | opelco 5882 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) | 
| 7 | df-br 5144 | . . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐶) | |
| 8 | 7 | bicomi 224 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝐶 ↔ 𝑥𝐶𝑧) | 
| 9 | 6, 8 | imbi12i 350 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| 10 | 19.23v 1942 | . . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
| 11 | 9, 10 | bitr4i 278 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| 12 | 11 | albii 1819 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| 13 | alcom 2159 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
| 14 | 12, 13 | bitri 275 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| 15 | 14 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝐴 ∘ 𝐵) → 〈𝑥, 𝑧〉 ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| 16 | 3, 15 | bitri 275 | 1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 ∘ 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-11 2157 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-co 5694 | 
| This theorem is referenced by: (None) | 
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