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 5873	. . 3 ⊢ ◡(𝑅 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑅)
2		cnvss 5856	. . 3 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → ◡(𝑅 ∘ 𝑆) ⊆ ◡𝑇)
3	1, 2	eqsstrrid 3973	. 2 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
4		cnvco 5873	. . . . 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 5856	. . . 4 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇)
9	7, 8	eqsstrrid 3973	. . 3 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → (𝑅 ∘ 𝑆) ⊆ ◡◡𝑇)
10		cnvcnvss 6191	. . 3 ⊢ ◡◡𝑇 ⊆ 𝑇
11	9, 10	sstrdi 3946	. 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 3902  ^◡ccnv 5658   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671

This theorem is used by: (None)