Theorem relcnvtrg 6116

Description: General form of relcnvtr 6117. (Contributed by Peter Mazsa, 17-Oct-2023.)

Assertion

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

Proof of Theorem relcnvtrg

Step	Hyp	Ref	Expression
1		cnvco 5753	. . 3 ⊢ ◡(𝑅 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑅)
2		cnvss 5740	. . 3 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → ◡(𝑅 ∘ 𝑆) ⊆ ◡𝑇)
3	1, 2	eqsstrrid 4013	. 2 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
4		cnvco 5753	. . . 4 ⊢ ◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆)
5		cnvss 5740	. . . 4 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇)
6		sseq1 3989	. . . . 5 ⊢ (◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆) → (◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇 ↔ (◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇))
7		dfrel2 6043	. . . . . . . . . 10 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8	7	biimpi 218	. . . . . . . . 9 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9	8	3ad2ant1 1128	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑅 = 𝑅)
10		dfrel2 6043	. . . . . . . . . 10 ⊢ (Rel 𝑆 ↔ ◡◡𝑆 = 𝑆)
11	10	biimpi 218	. . . . . . . . 9 ⊢ (Rel 𝑆 → ◡◡𝑆 = 𝑆)
12	11	3ad2ant2 1129	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑆 = 𝑆)
13	9, 12	coeq12d 5732	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (◡◡𝑅 ∘ ◡◡𝑆) = (𝑅 ∘ 𝑆))
14		dfrel2 6043	. . . . . . . . 9 ⊢ (Rel 𝑇 ↔ ◡◡𝑇 = 𝑇)
15	14	biimpi 218	. . . . . . . 8 ⊢ (Rel 𝑇 → ◡◡𝑇 = 𝑇)
16	15	3ad2ant3 1130	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑇 = 𝑇)
17	13, 16	sseq12d 3997	. . . . . 6 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇 ↔ (𝑅 ∘ 𝑆) ⊆ 𝑇))
18	17	biimpcd 251	. . . . 5 ⊢ ((◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇))
19	6, 18	syl6bi 255	. . . 4 ⊢ (◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆) → (◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇)))
20	4, 5, 19	mpsyl 68	. . 3 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇))
21	20	com12 32	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → (𝑅 ∘ 𝑆) ⊆ 𝑇))
22	3, 21	impbid2 228	1 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1082   = wceq 1536   ⊆ wss 3933  ^◡ccnv 5551   ∘ ccom 5556  Rel wrel 5557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pr 5327

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3495  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561

This theorem is referenced by:  relcnvtr  6117