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Theorem trrelsuperreldg 39891
Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 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 6114 . . . 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, 4sseqtrrd 4005 . 2 (𝜑𝑅𝑆)
6 xptrrel 14328 . . . 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 5728 . . 3 (𝜑 → (𝑆𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
97, 8, 43sstr4d 4011 . 2 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
105, 9jca 512 1 (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wss 3933   × cxp 5546  dom cdm 5548  ran crn 5549  ccom 5552  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by: (None)
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