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Theorem relcnvtrg 6093
 Description: General form of relcnvtr 6094. (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 5724 . . 3 (𝑅𝑆) = (𝑆𝑅)
2 cnvss 5711 . . 3 ((𝑅𝑆) ⊆ 𝑇(𝑅𝑆) ⊆ 𝑇)
31, 2eqsstrrid 3966 . 2 ((𝑅𝑆) ⊆ 𝑇 → (𝑆𝑅) ⊆ 𝑇)
4 cnvco 5724 . . . 4 (𝑆𝑅) = (𝑅𝑆)
5 cnvss 5711 . . . 4 ((𝑆𝑅) ⊆ 𝑇(𝑆𝑅) ⊆ 𝑇)
6 sseq1 3942 . . . . 5 ((𝑆𝑅) = (𝑅𝑆) → ((𝑆𝑅) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
7 dfrel2 6017 . . . . . . . . . 10 (Rel 𝑅𝑅 = 𝑅)
87biimpi 219 . . . . . . . . 9 (Rel 𝑅𝑅 = 𝑅)
983ad2ant1 1130 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑅 = 𝑅)
10 dfrel2 6017 . . . . . . . . . 10 (Rel 𝑆𝑆 = 𝑆)
1110biimpi 219 . . . . . . . . 9 (Rel 𝑆𝑆 = 𝑆)
12113ad2ant2 1131 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑆 = 𝑆)
139, 12coeq12d 5703 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) = (𝑅𝑆))
14 dfrel2 6017 . . . . . . . . 9 (Rel 𝑇𝑇 = 𝑇)
1514biimpi 219 . . . . . . . 8 (Rel 𝑇𝑇 = 𝑇)
16153ad2ant3 1132 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑇 = 𝑇)
1713, 16sseq12d 3950 . . . . . 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 1084   = wceq 1538   ⊆ wss 3883  ◡ccnv 5522   ∘ ccom 5527  Rel wrel 5528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532 This theorem is referenced by:  relcnvtr  6094
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