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Theorem rntrclfvOAI 42585
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 15052 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 rnss 5963 . . . 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 6176 . . . . 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 6202 . . . . 5 ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7 ssequn2 4206 . . . . 5 (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
86, 7mpbi 230 . . . 4 (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
95, 8eqtrdi 2790 . . 3 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
103, 9sseqtrd 4043 . 2 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11 trclfvlb 15053 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
12 rnss 5963 . . 3 (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
1311, 12syl 17 . 2 (𝑅𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
1410, 13eqssd 4020 1 (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  cun 3968  wss 3970   × cxp 5697  dom cdm 5699  ran crn 5700  cfv 6572  t+ctcl 15030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-iota 6524  df-fun 6574  df-fv 6580  df-trcl 15032
This theorem is referenced by: (None)
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