Theorem dmtrclfv 14961

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 14950	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		dmss 5900	. . . 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 5908	. . . 4 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5		dm0rn0 5922	. . . . . . 7 ⊢ (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6		xpeq1 5689	. . . . . . . . . 10 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7		0xp 5772	. . . . . . . . . 10 ⊢ (∅ × ran 𝑅) = ∅
8	6, 7	eqtrdi 2788	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
9	8	dmeqd 5903	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10		dm0 5918	. . . . . . . . 9 ⊢ dom ∅ = ∅
11	10	a1i 11	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom ∅ = ∅)
12		eqcom 2739	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
13	12	biimpi 215	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → ∅ = dom 𝑅)
14	9, 11, 13	3eqtrd 2776	. . . . . . 7 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
15	5, 14	sylbir 234	. . . . . 6 ⊢ (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16		dmxp 5926	. . . . . 6 ⊢ (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
17	15, 16	pm2.61ine 3025	. . . . 5 ⊢ dom (dom 𝑅 × ran 𝑅) = dom 𝑅
18	17	uneq2i 4159	. . . 4 ⊢ (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19		unidm 4151	. . . 4 ⊢ (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
20	4, 18, 19	3eqtri 2764	. . 3 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
21	3, 20	sseqtrdi 4031	. 2 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22		trclfvlb 14951	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
23		dmss 5900	. . 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 3998	1 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321   × cxp 5673  dom cdm 5675  ran crn 5676  ‘cfv 6540  t+ctcl 14928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-trcl 14930

This theorem is referenced by:  rntrclfvRP  42467