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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 6068 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6068. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2155. (Revised by BTernaryTau, 29-Dec-2024.) |
Ref | Expression |
---|---|
cotrg | ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 6064 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | ssrel3 5746 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧)) |
4 | vex 3451 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 3451 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
6 | 4, 5 | brco 5830 | . . . . . . 7 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) |
7 | 6 | imbi1i 350 | . . . . . 6 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
8 | 19.23v 1946 | . . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | |
9 | 7, 8 | bitr4i 278 | . . . . 5 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
10 | 9 | albii 1822 | . . . 4 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
11 | breq2 5113 | . . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑦𝐴𝑤)) | |
12 | 11 | anbi2d 630 | . . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤))) |
13 | breq2 5113 | . . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝑤)) | |
14 | 12, 13 | imbi12d 345 | . . . . 5 ⊢ (𝑧 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑤) → 𝑥𝐶𝑤))) |
15 | breq2 5113 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤)) | |
16 | breq1 5112 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧)) | |
17 | 15, 16 | anbi12d 632 | . . . . . 6 ⊢ (𝑦 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧))) |
18 | 17 | imbi1d 342 | . . . . 5 ⊢ (𝑦 = 𝑤 → (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ((𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧) → 𝑥𝐶𝑧))) |
19 | 14, 18 | alcomw 2048 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
20 | 10, 19 | bitri 275 | . . 3 ⊢ (∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
21 | 20 | albii 1822 | . 2 ⊢ (∀𝑥∀𝑧(𝑥(𝐴 ∘ 𝐵)𝑧 → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
22 | 3, 21 | bitri 275 | 1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 ⊆ wss 3914 class class class wbr 5109 ∘ ccom 5641 Rel wrel 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-co 5646 |
This theorem is referenced by: cotr 6068 dffun2 6510 dffun2OLD 6511 cotr2g 14870 |
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