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Theorem reltrclfv 14142
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 14132 . . . . 5 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
21adantr 474 . . . 4 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 simpr 479 . . . . 5 ((𝑅𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4 relssdmrn 5901 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5 ssequn1 4012 . . . . . 6 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
65biimpi 208 . . . . 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 3866 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9 xpss 5362 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (V × V)
108, 9syl6ss 3839 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11 df-rel 5353 . 2 (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
1210, 11sylibr 226 1 ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  cun 3796  wss 3798   × cxp 5344  dom cdm 5346  ran crn 5347  Rel wrel 5351  cfv 6127  t+ctcl 14110
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-8 2166  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-pow 5067  ax-pr 5129  ax-un 7214
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-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fv 6135  df-trcl 14112
This theorem is referenced by:  frege124d  38889  frege129d  38891  frege133d  38893
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