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Theorem rntrclfvOAI 36720
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 13677 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 rnss 5318 . . . 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 5504 . . . . 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 5529 . . . . 5 ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7 ssequn2 3769 . . . . 5 (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
86, 7mpbi 220 . . . 4 (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
95, 8syl6eq 2676 . . 3 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
103, 9sseqtrd 3625 . 2 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11 trclfvlb 13678 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
12 rnss 5318 . . 3 (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
1311, 12syl 17 . 2 (𝑅𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
1410, 13eqssd 3605 1 (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  cun 3558  wss 3560   × cxp 5077  dom cdm 5079  ran crn 5080  cfv 5850  t+ctcl 13653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5813  df-fun 5852  df-fv 5858  df-trcl 13655
This theorem is referenced by: (None)
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