Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trrelsuperrel2dg Structured version   Visualization version   GIF version

Theorem trrelsuperrel2dg 42422
Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
Hypothesis
Ref Expression
trrelsuperrel2dg.s (𝜑𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Assertion
Ref Expression
trrelsuperrel2dg (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Proof of Theorem trrelsuperrel2dg
StepHypRef Expression
1 ssun1 4173 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
2 trrelsuperrel2dg.s . . 3 (𝜑𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
31, 2sseqtrrid 4036 . 2 (𝜑𝑅𝑆)
4 xptrrel 14927 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
5 ssun2 4174 . . . . 5 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
64, 5sstri 3992 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
76a1i 11 . . 3 (𝜑 → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
82, 2coeq12d 5865 . . . 4 (𝜑 → (𝑆𝑆) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9 coundir 6248 . . . . . 6 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
10 relcnv 6104 . . . . . . 7 Rel 𝑅
11 cocnvcnv1 6257 . . . . . . . . 9 (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12 relssdmrn 6268 . . . . . . . . . . 11 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
13 dmcnvcnv 5933 . . . . . . . . . . . 12 dom 𝑅 = dom 𝑅
14 rncnvcnv 5934 . . . . . . . . . . . 12 ran 𝑅 = ran 𝑅
1513, 14xpeq12i 5705 . . . . . . . . . . 11 (dom 𝑅 × ran 𝑅) = (dom 𝑅 × ran 𝑅)
1612, 15sseqtrdi 4033 . . . . . . . . . 10 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
17 coss1 5856 . . . . . . . . . 10 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
1816, 17syl 17 . . . . . . . . 9 (Rel 𝑅 → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
1911, 18eqsstrrid 4032 . . . . . . . 8 (Rel 𝑅 → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
20 ssequn1 4181 . . . . . . . 8 ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ↔ ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
2119, 20sylib 217 . . . . . . 7 (Rel 𝑅 → ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
2210, 21ax-mp 5 . . . . . 6 ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
239, 22eqtri 2761 . . . . 5 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
24 coundi 6247 . . . . . 6 ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
25 cocnvcnv2 6258 . . . . . . . . 9 ((dom 𝑅 × ran 𝑅) ∘ 𝑅) = ((dom 𝑅 × ran 𝑅) ∘ 𝑅)
26 coss2 5857 . . . . . . . . . 10 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
2716, 26syl 17 . . . . . . . . 9 (Rel 𝑅 → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
2825, 27eqsstrrid 4032 . . . . . . . 8 (Rel 𝑅 → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29 ssequn1 4181 . . . . . . . 8 (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ↔ (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
3028, 29sylib 217 . . . . . . 7 (Rel 𝑅 → (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
3110, 30ax-mp 5 . . . . . 6 (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
3224, 31eqtri 2761 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
3323, 32eqtri 2761 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
348, 33eqtrdi 2789 . . 3 (𝜑 → (𝑆𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
357, 34, 23sstr4d 4030 . 2 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
363, 35jca 513 1 (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cun 3947  wss 3949   × cxp 5675  ccnv 5676  dom cdm 5677  ran crn 5678  ccom 5681  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator