Theorem dmtrclfv 15031

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 15020	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		dmss 5878	. . . 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 5886	. . . 4 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5		dm0rn0 5900	. . . . . . 7 ⊢ (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6		xpeq1 5661	. . . . . . . . . 10 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7		0xp 5746	. . . . . . . . . 10 ⊢ (∅ × ran 𝑅) = ∅
8	6, 7	eqtrdi 2813	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
9	8	dmeqd 5881	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10		dm0 5896	. . . . . . . . 9 ⊢ dom ∅ = ∅
11	10	a1i 11	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom ∅ = ∅)
12		eqcom 2769	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
13	12	biimpi 218	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → ∅ = dom 𝑅)
14	9, 11, 13	3eqtrd 2801	. . . . . . 7 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
15	5, 14	sylbir 237	. . . . . 6 ⊢ (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16		dmxp 5905	. . . . . 6 ⊢ (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
17	15, 16	pm2.61ine 3040	. . . . 5 ⊢ dom (dom 𝑅 × ran 𝑅) = dom 𝑅
18	17	uneq2i 4118	. . . 4 ⊢ (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19		unidm 4110	. . . 4 ⊢ (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
20	4, 18, 19	3eqtri 2789	. . 3 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
21	3, 20	sseqtrdi 3976	. 2 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22		trclfvlb 15021	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
23		dmss 5878	. . 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 3953	1 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285   × cxp 5645  dom cdm 5647  ran crn 5648  ‘cfv 6521  t+ctcl 14998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fv 6529  df-trcl 15000

This theorem is referenced by:  rntrclfvRP  44307