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Theorem reltrclfv 14827
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 14817 . . . . 5 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
21adantr 481 . . . 4 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 simpr 485 . . . . 5 ((𝑅𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4 relssdmrn 6206 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5 ssequn1 4127 . . . . . 6 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
65biimpi 215 . . . . 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 3972 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9 xpss 5636 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (V × V)
108, 9sstrdi 3944 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11 df-rel 5627 . 2 (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
1210, 11sylibr 233 1 ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  cun 3896  wss 3898   × cxp 5618  dom cdm 5620  ran crn 5621  Rel wrel 5625  cfv 6479  t+ctcl 14795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-iota 6431  df-fun 6481  df-fv 6487  df-trcl 14797
This theorem is referenced by:  frege124d  41690  frege129d  41692  frege133d  41694
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