Theorem cnvssb 40780

Description: Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
cnvssb	⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵))

Proof of Theorem cnvssb

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))

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)