Theorem relcnvtrgOLD 6269

Description: Obsolete form of relcnvtrg 6268 as of 8-Aug-2026. (Contributed by Peter Mazsa, 17-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem relcnvtrgOLD

Step	Hyp	Ref	Expression
1		cnvco 5866	. . 3 ⊢ ◡(𝑅 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑅)
2		cnvss 5849	. . 3 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → ◡(𝑅 ∘ 𝑆) ⊆ ◡𝑇)
3	1, 2	eqsstrrid 3947	. 2 ⊢ ((𝑅 ∘ 𝑆) ⊆ 𝑇 → (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
4		cnvco 5866	. . . 4 ⊢ ◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆)
5		cnvss 5849	. . . 4 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇)
6		sseq1 3933	. . . . 5 ⊢ (◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆) → (◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇 ↔ (◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇))
7		dfrel2 6180	. . . . . . . . . 10 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8	7	biimpi 219	. . . . . . . . 9 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅)
9	8	3ad2ant1 1151	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑅 = 𝑅)
10		dfrel2 6180	. . . . . . . . . 10 ⊢ (Rel 𝑆 ↔ ◡◡𝑆 = 𝑆)
11	10	biimpi 219	. . . . . . . . 9 ⊢ (Rel 𝑆 → ◡◡𝑆 = 𝑆)
12	11	3ad2ant2 1152	. . . . . . . 8 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑆 = 𝑆)
13	9, 12	coeq12d 5841	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (◡◡𝑅 ∘ ◡◡𝑆) = (𝑅 ∘ 𝑆))
14		dfrel2 6180	. . . . . . . . 9 ⊢ (Rel 𝑇 ↔ ◡◡𝑇 = 𝑇)
15	14	biimpi 219	. . . . . . . 8 ⊢ (Rel 𝑇 → ◡◡𝑇 = 𝑇)
16	15	3ad2ant3 1153	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ◡◡𝑇 = 𝑇)
17	13, 16	sseq12d 3941	. . . . . 6 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇 ↔ (𝑅 ∘ 𝑆) ⊆ 𝑇))
18	17	biimpcd 252	. . . . 5 ⊢ ((◡◡𝑅 ∘ ◡◡𝑆) ⊆ ◡◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇))
19	6, 18	biimtrdi 256	. . . 4 ⊢ (◡(◡𝑆 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑆) → (◡(◡𝑆 ∘ ◡𝑅) ⊆ ◡◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇)))
20	4, 5, 19	mpsyl 69	. . 3 ⊢ ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅 ∘ 𝑆) ⊆ 𝑇))
21	20	com12 33	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇 → (𝑅 ∘ 𝑆) ⊆ 𝑇))
22	3, 21	impbid2 229	1 ⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ⊆ wss 3895  ^◡ccnv 5649   ∘ ccom 5654  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659

This theorem is used by:  relcnvtrOLD  6270