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Theorem relcnvtrg 6264
Description: General form of relcnvtr 6265. (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 5884 . . 3 (𝑅𝑆) = (𝑆𝑅)
2 cnvss 5871 . . 3 ((𝑅𝑆) ⊆ 𝑇(𝑅𝑆) ⊆ 𝑇)
31, 2eqsstrrid 4030 . 2 ((𝑅𝑆) ⊆ 𝑇 → (𝑆𝑅) ⊆ 𝑇)
4 cnvco 5884 . . . 4 (𝑆𝑅) = (𝑅𝑆)
5 cnvss 5871 . . . 4 ((𝑆𝑅) ⊆ 𝑇(𝑆𝑅) ⊆ 𝑇)
6 sseq1 4006 . . . . 5 ((𝑆𝑅) = (𝑅𝑆) → ((𝑆𝑅) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
7 dfrel2 6187 . . . . . . . . . 10 (Rel 𝑅𝑅 = 𝑅)
87biimpi 215 . . . . . . . . 9 (Rel 𝑅𝑅 = 𝑅)
983ad2ant1 1131 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑅 = 𝑅)
10 dfrel2 6187 . . . . . . . . . 10 (Rel 𝑆𝑆 = 𝑆)
1110biimpi 215 . . . . . . . . 9 (Rel 𝑆𝑆 = 𝑆)
12113ad2ant2 1132 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑆 = 𝑆)
139, 12coeq12d 5863 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) = (𝑅𝑆))
14 dfrel2 6187 . . . . . . . . 9 (Rel 𝑇𝑇 = 𝑇)
1514biimpi 215 . . . . . . . 8 (Rel 𝑇𝑇 = 𝑇)
16153ad2ant3 1133 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑇 = 𝑇)
1713, 16sseq12d 4014 . . . . . 6 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
1817biimpcd 248 . . . . 5 ((𝑅𝑆) ⊆ 𝑇 → ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) ⊆ 𝑇))
196, 18syl6bi 252 . . . 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 225 1 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑆𝑅) ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1539  wss 3947  ccnv 5674  ccom 5679  Rel wrel 5680
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684
This theorem is referenced by:  relcnvtr  6265
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