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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 5745 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | cnvss 5745 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
3 | dfrel2 6048 | . . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
4 | 3 | biimpi 218 | . . . . . . 7 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
5 | 4 | eqcomd 2829 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 = ◡◡𝐴) |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 = ◡◡𝐴) |
7 | id 22 | . . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
8 | cnvcnvss 6053 | . . . . . . 7 ⊢ ◡◡𝐵 ⊆ 𝐵 | |
9 | 7, 8 | sstrdi 3981 | . . . . . 6 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ 𝐵) |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → ◡◡𝐴 ⊆ 𝐵) |
11 | 6, 10 | eqsstrd 4007 | . . . 4 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 ⊆ 𝐵) |
12 | 11 | ex 415 | . . 3 ⊢ (Rel 𝐴 → (◡◡𝐴 ⊆ ◡◡𝐵 → 𝐴 ⊆ 𝐵)) |
13 | 2, 12 | syl5 34 | . 2 ⊢ (Rel 𝐴 → (◡𝐴 ⊆ ◡𝐵 → 𝐴 ⊆ 𝐵)) |
14 | 1, 13 | impbid2 228 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ⊆ wss 3938 ◡ccnv 5556 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: (None) |
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