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Theorem trrelsuperreldg 38486
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 5799 . . . 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 3788 . 2 (𝜑𝑅𝑆)
6 xptrrel 13932 . . . 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 5424 . . 3 (𝜑 → (𝑆𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
97, 8, 43sstr4d 3794 . 2 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
105, 9jca 556 1 (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wss 3720   × cxp 5246  dom cdm 5248  ran crn 5249  ccom 5252  Rel wrel 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4784  df-opab 4844  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260
This theorem is referenced by: (None)
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