Theorem dmtrclfv 14380

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 14369	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		dmss 5773	. . . 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 5781	. . . 4 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5		dm0rn0 5797	. . . . . . 7 ⊢ (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6		xpeq1 5571	. . . . . . . . . 10 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7		0xp 5651	. . . . . . . . . 10 ⊢ (∅ × ran 𝑅) = ∅
8	6, 7	syl6eq 2874	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
9	8	dmeqd 5776	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10		dm0 5792	. . . . . . . . 9 ⊢ dom ∅ = ∅
11	10	a1i 11	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom ∅ = ∅)
12		eqcom 2830	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
13	12	biimpi 218	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → ∅ = dom 𝑅)
14	9, 11, 13	3eqtrd 2862	. . . . . . 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 5801	. . . . . 6 ⊢ (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
17	15, 16	pm2.61ine 3102	. . . . 5 ⊢ dom (dom 𝑅 × ran 𝑅) = dom 𝑅
18	17	uneq2i 4138	. . . 4 ⊢ (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19		unidm 4130	. . . 4 ⊢ (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
20	4, 18, 19	3eqtri 2850	. . 3 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
21	3, 20	sseqtrdi 4019	. 2 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22		trclfvlb 14370	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
23		dmss 5773	. . 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 3986	1 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293   × cxp 5555  dom cdm 5557  ran crn 5558  ‘cfv 6357  t+ctcl 14347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-trcl 14349

This theorem is referenced by:  rntrclfvRP  40083