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Theorem reltrclfv 14365
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 14355 . . . . 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 6114 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5 ssequn1 4153 . . . . . 6 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
65biimpi 217 . . . . 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 4004 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9 xpss 5564 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (V × V)
108, 9sstrdi 3976 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11 df-rel 5555 . 2 (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
1210, 11sylibr 235 1 ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  wss 3933   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553  cfv 6348  t+ctcl 14333
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-pow 5257  ax-pr 5320  ax-un 7450
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-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-trcl 14335
This theorem is referenced by:  frege124d  39984  frege129d  39986  frege133d  39988
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