Theorem relcnvtrg 6119

Description: General form of relcnvtr 6120. (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 5756	. . 3 ⊢ ◡(𝑅 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑅)
2		cnvss 5743	. . 3 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → ◡(𝑅 ∘ 𝑆) ⊆ ◡𝑇)
3	1, 2	eqsstrrid 4016	. 2 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
4		cnvco 5756	. . . 4 ⊢ ◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆)
5		cnvss 5743	. . . 4 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇)
6		sseq1 3992	. . . . 5 ⊢ (◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆) → (◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇 ↔ (◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇))
7		dfrel2 6046	. . . . . . . . . 10 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8	7	biimpi 218	. . . . . . . . 9 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9	8	3ad2ant1 1129	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑅 = 𝑅)
10		dfrel2 6046	. . . . . . . . . 10 ⊢ (Rel 𝑆 ↔ ◡◡𝑆 = 𝑆)
11	10	biimpi 218	. . . . . . . . 9 ⊢ (Rel 𝑆 → ◡◡𝑆 = 𝑆)
12	11	3ad2ant2 1130	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑆 = 𝑆)
13	9, 12	coeq12d 5735	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (◡◡𝑅 ∘ ◡◡𝑆) = (𝑅 ∘ 𝑆))
14		dfrel2 6046	. . . . . . . . 9 ⊢ (Rel 𝑇 ↔ ◡◡𝑇 = 𝑇)
15	14	biimpi 218	. . . . . . . 8 ⊢ (Rel 𝑇 → ◡◡𝑇 = 𝑇)
16	15	3ad2ant3 1131	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑇 = 𝑇)
17	13, 16	sseq12d 4000	. . . . . 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 1083   = wceq 1537   ⊆ wss 3936  ^◡ccnv 5554   ∘ ccom 5559  Rel wrel 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564

This theorem is referenced by:  relcnvtr  6120