Theorem rntrclfvOAI 43140

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 14960	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		rnss 5881	. . . 4 ⊢ ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4		rnun 6096	. . . . 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 6123	. . . . 5 ⊢ ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7		ssequn2 4118	. . . . 5 ⊢ (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
8	6, 7	mpbi 231	. . . 4 ⊢ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
9	5, 8	eqtrdi 2790	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
10	3, 9	sseqtrd 3951	. 2 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11		trclfvlb 14961	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
12		rnss 5881	. . 3 ⊢ (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
13	11, 12	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
14	10, 13	eqssd 3932	1 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∪ cun 3881   ⊆ wss 3883   × cxp 5616  dom cdm 5618  ran crn 5619  ‘cfv 6485  t+ctcl 14938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-trcl 14940

This theorem is referenced by: (None)