Theorem dmtrclfv 15042

Description: The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
dmtrclfv	⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Proof of Theorem dmtrclfv

Step	Hyp	Ref	Expression
1		trclfvub 15031	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		dmss 5887	. . . 4 ⊢ ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → dom (t+‘𝑅) ⊆ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4		dmun 5895	. . . 4 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5		dm0rn0 5909	. . . . . . 7 ⊢ (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6		xpeq1 5673	. . . . . . . . . 10 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7		0xp 5758	. . . . . . . . . 10 ⊢ (∅ × ran 𝑅) = ∅
8	6, 7	eqtrdi 2787	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
9	8	dmeqd 5890	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10		dm0 5905	. . . . . . . . 9 ⊢ dom ∅ = ∅
11	10	a1i 11	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom ∅ = ∅)
12		eqcom 2743	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
13	12	biimpi 216	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → ∅ = dom 𝑅)
14	9, 11, 13	3eqtrd 2775	. . . . . . 7 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
15	5, 14	sylbir 235	. . . . . 6 ⊢ (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16		dmxp 5913	. . . . . 6 ⊢ (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
17	15, 16	pm2.61ine 3016	. . . . 5 ⊢ dom (dom 𝑅 × ran 𝑅) = dom 𝑅
18	17	uneq2i 4145	. . . 4 ⊢ (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19		unidm 4137	. . . 4 ⊢ (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
20	4, 18, 19	3eqtri 2763	. . 3 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
21	3, 20	sseqtrdi 4004	. 2 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22		trclfvlb 15032	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
23		dmss 5887	. . 3 ⊢ (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
24	22, 23	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
25	21, 24	eqssd 3981	1 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313   × cxp 5657  dom cdm 5659  ran crn 5660  ‘cfv 6536  t+ctcl 15009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  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 6538  df-fv 6544  df-trcl 15011

This theorem is referenced by:  rntrclfvRP  43722