Theorem cnvssb 38675

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 5498	. 2 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
2		cnvss 5498	. . 3 ⊢ (◡𝐴 ⊆ ◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵)
3		dfrel2 5800	. . . . . . . 8 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
4	3	biimpi 208	. . . . . . 7 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴)
5	4	eqcomd 2805	. . . . . 6 ⊢ (Rel 𝐴 → 𝐴 = ◡◡𝐴)
6	5	adantr 473	. . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 = ◡◡𝐴)
7		id 22	. . . . . . 7 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ ◡◡𝐵)
8		cnvcnvss 5805	. . . . . . 7 ⊢ ◡◡𝐵 ⊆ 𝐵
9	7, 8	syl6ss 3810	. . . . . 6 ⊢ (◡◡𝐴 ⊆ ◡◡𝐵 → ◡◡𝐴 ⊆ 𝐵)
10	9	adantl 474	. . . . 5 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → ◡◡𝐴 ⊆ 𝐵)
11	6, 10	eqsstrd 3835	. . . 4 ⊢ ((Rel 𝐴 ∧ ◡◡𝐴 ⊆ ◡◡𝐵) → 𝐴 ⊆ 𝐵)
12	11	ex 402	. . 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))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ⊆ wss 3769  ^◡ccnv 5311  Rel wrel 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320

This theorem is referenced by: (None)