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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 5725 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | cnvss 5725 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
3 | dfrel2 6031 | . . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
4 | 3 | biimpi 219 | . . . . . . 7 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
5 | 4 | eqcomd 2745 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 = ◡◡𝐴) |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 = ◡◡𝐴) |
7 | id 22 | . . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
8 | cnvcnvss 6036 | . . . . . . 7 ⊢ ◡◡𝐵 ⊆ 𝐵 | |
9 | 7, 8 | sstrdi 3899 | . . . . . 6 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ 𝐵) |
10 | 9 | adantl 485 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → ◡◡𝐴 ⊆ 𝐵) |
11 | 6, 10 | eqsstrd 3925 | . . . 4 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 ⊆ 𝐵) |
12 | 11 | ex 416 | . . 3 ⊢ (Rel 𝐴 → (◡◡𝐴 ⊆ ◡◡𝐵 → 𝐴 ⊆ 𝐵)) |
13 | 2, 12 | syl5 34 | . 2 ⊢ (Rel 𝐴 → (◡𝐴 ⊆ ◡𝐵 → 𝐴 ⊆ 𝐵)) |
14 | 1, 13 | impbid2 229 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ⊆ wss 3853 ◡ccnv 5534 Rel wrel 5540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-rel 5542 df-cnv 5543 |
This theorem is referenced by: (None) |
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