Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trrelsuperreldg Structured version   Visualization version   GIF version

Theorem trrelsuperreldg 38796
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 5901 . . . 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 3867 . 2 (𝜑𝑅𝑆)
6 xptrrel 14105 . . . 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 5523 . . 3 (𝜑 → (𝑆𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
97, 8, 43sstr4d 3873 . 2 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
105, 9jca 507 1 (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wss 3798   × cxp 5344  dom cdm 5346  ran crn 5347  ccom 5350  Rel wrel 5351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator