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Theorem relcnvtrg 6130
Description: General form of relcnvtr 6131. (Contributed by Peter Mazsa, 17-Oct-2023.)
Assertion
Ref Expression
relcnvtrg ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑆𝑅) ⊆ 𝑇))

Proof of Theorem relcnvtrg
StepHypRef Expression
1 cnvco 5754 . . 3 (𝑅𝑆) = (𝑆𝑅)
2 cnvss 5741 . . 3 ((𝑅𝑆) ⊆ 𝑇(𝑅𝑆) ⊆ 𝑇)
31, 2eqsstrrid 3950 . 2 ((𝑅𝑆) ⊆ 𝑇 → (𝑆𝑅) ⊆ 𝑇)
4 cnvco 5754 . . . 4 (𝑆𝑅) = (𝑅𝑆)
5 cnvss 5741 . . . 4 ((𝑆𝑅) ⊆ 𝑇(𝑆𝑅) ⊆ 𝑇)
6 sseq1 3926 . . . . 5 ((𝑆𝑅) = (𝑅𝑆) → ((𝑆𝑅) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
7 dfrel2 6052 . . . . . . . . . 10 (Rel 𝑅𝑅 = 𝑅)
87biimpi 219 . . . . . . . . 9 (Rel 𝑅𝑅 = 𝑅)
983ad2ant1 1135 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑅 = 𝑅)
10 dfrel2 6052 . . . . . . . . . 10 (Rel 𝑆𝑆 = 𝑆)
1110biimpi 219 . . . . . . . . 9 (Rel 𝑆𝑆 = 𝑆)
12113ad2ant2 1136 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑆 = 𝑆)
139, 12coeq12d 5733 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) = (𝑅𝑆))
14 dfrel2 6052 . . . . . . . . 9 (Rel 𝑇𝑇 = 𝑇)
1514biimpi 219 . . . . . . . 8 (Rel 𝑇𝑇 = 𝑇)
16153ad2ant3 1137 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑇 = 𝑇)
1713, 16sseq12d 3934 . . . . . 6 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
1817biimpcd 252 . . . . 5 ((𝑅𝑆) ⊆ 𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) ⊆ 𝑇))
196, 18syl6bi 256 . . . 4 ((𝑆𝑅) = (𝑅𝑆) → ((𝑆𝑅) ⊆ 𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) ⊆ 𝑇)))
204, 5, 19mpsyl 68 . . 3 ((𝑆𝑅) ⊆ 𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) ⊆ 𝑇))
2120com12 32 . 2 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑆𝑅) ⊆ 𝑇 → (𝑅𝑆) ⊆ 𝑇))
223, 21impbid2 229 1 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑆𝑅) ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wss 3866  ccnv 5550  ccom 5555  Rel wrel 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560
This theorem is referenced by:  relcnvtr  6131
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