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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvssb | Structured version Visualization version GIF version |
Description: Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.) |
Ref | Expression |
---|---|
cnvssb | ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5781 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | cnvss 5781 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
3 | dfrel2 6092 | . . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
4 | 3 | biimpi 215 | . . . . . . 7 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
5 | 4 | eqcomd 2744 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 = ◡◡𝐴) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 = ◡◡𝐴) |
7 | id 22 | . . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
8 | cnvcnvss 6097 | . . . . . . 7 ⊢ ◡◡𝐵 ⊆ 𝐵 | |
9 | 7, 8 | sstrdi 3933 | . . . . . 6 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ 𝐵) |
10 | 9 | adantl 482 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → ◡◡𝐴 ⊆ 𝐵) |
11 | 6, 10 | eqsstrd 3959 | . . . 4 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 ⊆ 𝐵) |
12 | 11 | ex 413 | . . 3 ⊢ (Rel 𝐴 → (◡◡𝐴 ⊆ ◡◡𝐵 → 𝐴 ⊆ 𝐵)) |
13 | 2, 12 | syl5 34 | . 2 ⊢ (Rel 𝐴 → (◡𝐴 ⊆ ◡𝐵 → 𝐴 ⊆ 𝐵)) |
14 | 1, 13 | impbid2 225 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ⊆ wss 3887 ◡ccnv 5588 Rel wrel 5594 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 |
This theorem is referenced by: (None) |
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