Theorem dmtrclfv 14941

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 14930	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		dmss 5851	. . . 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 5859	. . . 4 ⊢ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5		dm0rn0 5873	. . . . . . 7 ⊢ (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6		xpeq1 5638	. . . . . . . . . 10 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7		0xp 5723	. . . . . . . . . 10 ⊢ (∅ × ran 𝑅) = ∅
8	6, 7	eqtrdi 2787	. . . . . . . . 9 ⊢ (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
9	8	dmeqd 5854	. . . . . . . 8 ⊢ (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10		dm0 5869	. . . . . . . . 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 5878	. . . . . 6 ⊢ (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
17	15, 16	pm2.61ine 3015	. . . . 5 ⊢ dom (dom 𝑅 × ran 𝑅) = dom 𝑅
18	17	uneq2i 4117	. . . 4 ⊢ (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19		unidm 4109	. . . 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 3974	. 2 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22		trclfvlb 14931	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
23		dmss 5851	. . 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 3951	1 ⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285   × cxp 5622  dom cdm 5624  ran crn 5625  ‘cfv 6492  t+ctcl 14908

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-trcl 14910

This theorem is referenced by:  rntrclfvRP  43972