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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 5527 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | cnvss 5527 | . . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
3 | dfrel2 5824 | . . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
4 | 3 | biimpi 208 | . . . . . . 7 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
5 | 4 | eqcomd 2831 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 = ◡◡𝐴) |
6 | 5 | adantr 474 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 = ◡◡𝐴) |
7 | id 22 | . . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵) | |
8 | cnvcnvss 5829 | . . . . . . 7 ⊢ ◡◡𝐵 ⊆ 𝐵 | |
9 | 7, 8 | syl6ss 3839 | . . . . . 6 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ 𝐵) |
10 | 9 | adantl 475 | . . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → ◡◡𝐴 ⊆ 𝐵) |
11 | 6, 10 | eqsstrd 3864 | . . . 4 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 ⊆ 𝐵) |
12 | 11 | ex 403 | . . 3 ⊢ (Rel 𝐴 → (◡◡𝐴 ⊆ ◡◡𝐵 → 𝐴 ⊆ 𝐵)) |
13 | 2, 12 | syl5 34 | . 2 ⊢ (Rel 𝐴 → (◡𝐴 ⊆ ◡𝐵 → 𝐴 ⊆ 𝐵)) |
14 | 1, 13 | impbid2 218 | 1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ⊆ wss 3798 ◡ccnv 5341 Rel wrel 5347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 |
This theorem is referenced by: (None) |
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