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Theorem rntrclfvOAI 40917
Description: The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
Assertion
Ref Expression
rntrclfvOAI (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

Proof of Theorem rntrclfvOAI
StepHypRef Expression
1 trclfvub 14846 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 rnss 5892 . . . 4 ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
31, 2syl 17 . . 3 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4 rnun 6096 . . . . 5 ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅))
54a1i 11 . . . 4 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)))
6 rnxpss 6122 . . . . 5 ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7 ssequn2 4141 . . . . 5 (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
86, 7mpbi 229 . . . 4 (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
95, 8eqtrdi 2792 . . 3 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
103, 9sseqtrd 3982 . 2 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11 trclfvlb 14847 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
12 rnss 5892 . . 3 (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
1311, 12syl 17 . 2 (𝑅𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
1410, 13eqssd 3959 1 (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cun 3906  wss 3908   × cxp 5629  dom cdm 5631  ran crn 5632  cfv 6493  t+ctcl 14824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501  df-trcl 14826
This theorem is referenced by: (None)
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