Theorem relcnvtrg 6267

Description: Subclass law for converse of a composition. A particular case is ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (^◡𝑅 ∘ ^◡𝑅) ⊆ ^◡𝑅), which says that a relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) Generalize to a statement about three classes rather than one. (Revised by Peter Mazsa, 17-Oct-2023.) Remove antecedent. (Revised by Eric Schmidt, 8-Aug-2026.)

Assertion

Ref	Expression
relcnvtrg	⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)

Proof of Theorem relcnvtrg

Step	Hyp	Ref	Expression
1		cnvco 5874	. . 3 ⊢ ◡(𝑅 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑅)
2		cnvss 5857	. . 3 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → ◡(𝑅 ∘ 𝑆) ⊆ ◡𝑇)
3	1, 2	eqsstrrid 3975	. 2 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
4		cnvco 5874	. . . . 5 ⊢ ◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆)
5		cocnvcnv1 6258	. . . . 5 ⊢ (◡◡𝑅 ∘ ◡◡𝑆) = (𝑅 ∘ ◡◡𝑆)
6		cocnvcnv2 6259	. . . . 5 ⊢ (𝑅 ∘ ◡◡𝑆) = (𝑅 ∘ 𝑆)
7	4, 5, 6	3eqtri 2789	. . . 4 ⊢ ◡(◡𝑆 ∘ ◡𝑅) = (𝑅 ∘ 𝑆)
8		cnvss 5857	. . . 4 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇)
9	7, 8	eqsstrrid 3975	. . 3 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → (𝑅 ∘ 𝑆) ⊆ ◡◡𝑇)
10		cnvcnvss 6191	. . 3 ⊢ ◡◡𝑇 ⊆ 𝑇
11	9, 10	sstrdi 3948	. 2 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → (𝑅 ∘ 𝑆) ⊆ 𝑇)
12	3, 11	impbii 212	1 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)

This proof depends on syntax axioms:   ↔ wb 209   ⊆ wss 3904  ^◡ccnv 5659   ∘ ccom 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672

This theorem is used by: (None)