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Theorem trrelsuperreldg 37476
Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
Hypotheses
Ref Expression
trrelsuperreldg.r (𝜑 → Rel 𝑅)
trrelsuperreldg.s (𝜑𝑆 = (dom 𝑅 × ran 𝑅))
Assertion
Ref Expression
trrelsuperreldg (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Proof of Theorem trrelsuperreldg
StepHypRef Expression
1 trrelsuperreldg.r . . . 4 (𝜑 → Rel 𝑅)
2 relssdmrn 5620 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 17 . . 3 (𝜑𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 trrelsuperreldg.s . . 3 (𝜑𝑆 = (dom 𝑅 × ran 𝑅))
53, 4sseqtr4d 3626 . 2 (𝜑𝑅𝑆)
6 xptrrel 13661 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
76a1i 11 . . 3 (𝜑 → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅))
84, 4coeq12d 5251 . . 3 (𝜑 → (𝑆𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
97, 8, 43sstr4d 3632 . 2 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
105, 9jca 554 1 (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wss 3559   × cxp 5077  dom cdm 5079  ran crn 5080  ccom 5083  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091
This theorem is referenced by: (None)
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