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Theorem relcnvtrg 6218
Description: General form of relcnvtr 6219. (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 5841 . . 3 (𝑅𝑆) = (𝑆𝑅)
2 cnvss 5828 . . 3 ((𝑅𝑆) ⊆ 𝑇(𝑅𝑆) ⊆ 𝑇)
31, 2eqsstrrid 3993 . 2 ((𝑅𝑆) ⊆ 𝑇 → (𝑆𝑅) ⊆ 𝑇)
4 cnvco 5841 . . . 4 (𝑆𝑅) = (𝑅𝑆)
5 cnvss 5828 . . . 4 ((𝑆𝑅) ⊆ 𝑇(𝑆𝑅) ⊆ 𝑇)
6 sseq1 3969 . . . . 5 ((𝑆𝑅) = (𝑅𝑆) → ((𝑆𝑅) ⊆ 𝑇 ↔ (𝑅𝑆) ⊆ 𝑇))
7 dfrel2 6141 . . . . . . . . . 10 (Rel 𝑅𝑅 = 𝑅)
87biimpi 215 . . . . . . . . 9 (Rel 𝑅𝑅 = 𝑅)
983ad2ant1 1133 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑅 = 𝑅)
10 dfrel2 6141 . . . . . . . . . 10 (Rel 𝑆𝑆 = 𝑆)
1110biimpi 215 . . . . . . . . 9 (Rel 𝑆𝑆 = 𝑆)
12113ad2ant2 1134 . . . . . . . 8 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑆 = 𝑆)
139, 12coeq12d 5820 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → (𝑅𝑆) = (𝑅𝑆))
14 dfrel2 6141 . . . . . . . . 9 (Rel 𝑇𝑇 = 𝑇)
1514biimpi 215 . . . . . . . 8 (Rel 𝑇𝑇 = 𝑇)
16153ad2ant3 1135 . . . . . . 7 ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → 𝑇 = 𝑇)
1713, 16sseq12d 3977 . . . . . 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 1087   = wceq 1541  wss 3910  ccnv 5632  ccom 5637  Rel wrel 5638
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642
This theorem is referenced by:  relcnvtr  6219
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