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Theorem reltrclfv 15057
Description: The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
reltrclfv ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Proof of Theorem reltrclfv
StepHypRef Expression
1 trclfvub 15047 . . . . 5 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
21adantr 480 . . . 4 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 simpr 484 . . . . 5 ((𝑅𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4 relssdmrn 6287 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5 ssequn1 4185 . . . . . 6 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
65biimpi 216 . . . . 5 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
73, 4, 63syl 18 . . . 4 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
82, 7sseqtrd 4019 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9 xpss 5700 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (V × V)
108, 9sstrdi 3995 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11 df-rel 5691 . 2 (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
1210, 11sylibr 234 1 ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  wss 3950   × cxp 5682  dom cdm 5684  ran crn 5685  Rel wrel 5689  cfv 6560  t+ctcl 15025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-trcl 15027
This theorem is referenced by:  frege124d  43779  frege129d  43781  frege133d  43783
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