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Theorem reltrclfv 13966
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 13956 . . . . 5 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
21adantr 466 . . . 4 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 simpr 471 . . . . 5 ((𝑅𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4 relssdmrn 5799 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5 ssequn1 3934 . . . . . 6 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
65biimpi 206 . . . . 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 3790 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9 xpss 5266 . . 3 (dom 𝑅 × ran 𝑅) ⊆ (V × V)
108, 9syl6ss 3764 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11 df-rel 5257 . 2 (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
1210, 11sylibr 224 1 ((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  wss 3723   × cxp 5248  dom cdm 5250  ran crn 5251  Rel wrel 5255  cfv 6030  t+ctcl 13934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-trcl 13936
This theorem is referenced by:  frege124d  38577  frege129d  38579  frege133d  38581
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