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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvssco | Structured version Visualization version GIF version |
Description: A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.) |
Ref | Expression |
---|---|
cnvssco | ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcom 2153 | . 2 ⊢ (∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶)) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶))) | |
2 | relcnv 5960 | . . 3 ⊢ Rel ◡𝐴 | |
3 | ssrel 5650 | . . 3 ⊢ (Rel ◡𝐴 → (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶)))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑦∀𝑥(〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶))) |
5 | 19.37v 1989 | . . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))) | |
6 | vex 3495 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | 6, 7 | brcnv 5746 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
9 | df-br 5058 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝐴) | |
10 | 8, 9 | bitr3i 278 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝐴) |
11 | 7, 6 | brco 5734 | . . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) |
12 | 6, 7 | brcnv 5746 | . . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ 𝑥(𝐵 ∘ 𝐶)𝑦) |
13 | df-br 5058 | . . . . . . 7 ⊢ (𝑦◡(𝐵 ∘ 𝐶)𝑥 ↔ 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶)) | |
14 | 12, 13 | bitr3i 278 | . . . . . 6 ⊢ (𝑥(𝐵 ∘ 𝐶)𝑦 ↔ 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶)) |
15 | 11, 14 | bitr3i 278 | . . . . 5 ⊢ (∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦) ↔ 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶)) |
16 | 10, 15 | imbi12i 352 | . . . 4 ⊢ ((𝑥𝐴𝑦 → ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶))) |
17 | 5, 16 | bitri 276 | . . 3 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ (〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶))) |
18 | 17 | 2albii 1812 | . 2 ⊢ (∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)) ↔ ∀𝑥∀𝑦(〈𝑦, 𝑥〉 ∈ ◡𝐴 → 〈𝑦, 𝑥〉 ∈ ◡(𝐵 ∘ 𝐶))) |
19 | 1, 4, 18 | 3bitr4i 304 | 1 ⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∈ wcel 2105 ⊆ wss 3933 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 ∘ ccom 5552 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 |
This theorem is referenced by: refimssco 39845 |
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