Theorem rntrclfvOAI 43307

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	⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)

Proof of Theorem rntrclfvOAI

Step	Hyp	Ref	Expression
1		trclfvub 15040	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		rnss 5927	. . . 4 ⊢ ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	syl 18	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4		rnun 6140	. . . . 5 ⊢ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅))
5	4	a1i 11	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)))
6		rnxpss 6169	. . . . 5 ⊢ ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7		ssequn2 4150	. . . . 5 ⊢ (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
8	6, 7	mpbi 233	. . . 4 ⊢ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
9	5, 8	eqtrdi 2820	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
10	3, 9	sseqtrd 3981	. 2 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11		trclfvlb 15041	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
12		rnss 5927	. . 3 ⊢ (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
13	11, 12	syl 18	. 2 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
14	10, 13	eqssd 3962	1 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ∪ cun 3911   ⊆ wss 3913   × cxp 5657  dom cdm 5659  ran crn 5660  ‘cfv 6533  t+ctcl 15018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fv 6541  df-trcl 15020

This theorem is referenced by: (None)