Users' Mathboxes Mathbox for OpenAI < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rntrclfvOAI Structured version   Visualization version   GIF version

Theorem rntrclfvOAI 41061
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 14901 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 rnss 5898 . . . 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 6102 . . . . 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 6128 . . . . 5 ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7 ssequn2 4147 . . . . 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 2789 . . 3 (𝑅𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
103, 9sseqtrd 3988 . 2 (𝑅𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11 trclfvlb 14902 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
12 rnss 5898 . . 3 (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
1311, 12syl 17 . 2 (𝑅𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
1410, 13eqssd 3965 1 (𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cun 3912  wss 3914   × cxp 5635  dom cdm 5637  ran crn 5638  cfv 6500  t+ctcl 14879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-trcl 14881
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator