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Mirrors > Home > MPE Home > Th. List > cnvsymOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of cnvsym 6107 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnvsymOLD | ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6097 | . . 3 ⊢ Rel ◡𝑅 | |
2 | ssrel3 5779 | . . 3 ⊢ (Rel ◡𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)) |
4 | alcom 2148 | . 2 ⊢ (∀𝑦∀𝑥(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥)) | |
5 | vex 3472 | . . . . 5 ⊢ 𝑦 ∈ V | |
6 | vex 3472 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 5876 | . . . 4 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
8 | 7 | imbi1i 349 | . . 3 ⊢ ((𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
9 | 8 | 2albii 1814 | . 2 ⊢ (∀𝑥∀𝑦(𝑦◡𝑅𝑥 → 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
10 | 3, 4, 9 | 3bitri 297 | 1 ⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ⊆ wss 3943 class class class wbr 5141 ◡ccnv 5668 Rel wrel 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 |
This theorem is referenced by: (None) |
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