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Theorem rntrclfvOAI 42649
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 15058 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 rnss 5964 . . . 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 6179 . . . . 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 6205 . . . . 5 ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7 ssequn2 4212 . . . . 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 2796 . . 3 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
103, 9sseqtrd 4049 . 2 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11 trclfvlb 15059 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
12 rnss 5964 . . 3 (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
1311, 12syl 17 . 2 (𝑅𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
1410, 13eqssd 4026 1 (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  cun 3974  wss 3976   × cxp 5698  dom cdm 5700  ran crn 5701  cfv 6575  t+ctcl 15036
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6527  df-fun 6577  df-fv 6583  df-trcl 15038
This theorem is referenced by: (None)
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