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Theorem dmtrclfv 15057
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 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)

Proof of Theorem dmtrclfv
StepHypRef Expression
1 trclfvub 15046 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 dmss 5913 . . . 4 ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → dom (t+‘𝑅) ⊆ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
31, 2syl 17 . . 3 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4 dmun 5921 . . . 4 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5 dm0rn0 5935 . . . . . . 7 (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6 xpeq1 5699 . . . . . . . . . 10 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7 0xp 5784 . . . . . . . . . 10 (∅ × ran 𝑅) = ∅
86, 7eqtrdi 2793 . . . . . . . . 9 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
98dmeqd 5916 . . . . . . . 8 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10 dm0 5931 . . . . . . . . 9 dom ∅ = ∅
1110a1i 11 . . . . . . . 8 (dom 𝑅 = ∅ → dom ∅ = ∅)
12 eqcom 2744 . . . . . . . . 9 (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
1312biimpi 216 . . . . . . . 8 (dom 𝑅 = ∅ → ∅ = dom 𝑅)
149, 11, 133eqtrd 2781 . . . . . . 7 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
155, 14sylbir 235 . . . . . 6 (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16 dmxp 5939 . . . . . 6 (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
1715, 16pm2.61ine 3025 . . . . 5 dom (dom 𝑅 × ran 𝑅) = dom 𝑅
1817uneq2i 4165 . . . 4 (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19 unidm 4157 . . . 4 (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
204, 18, 193eqtri 2769 . . 3 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
213, 20sseqtrdi 4024 . 2 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22 trclfvlb 15047 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
23 dmss 5913 . . 3 (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
2422, 23syl 17 . 2 (𝑅𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
2521, 24eqssd 4001 1 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cun 3949  wss 3951  c0 4333   × cxp 5683  dom cdm 5685  ran crn 5686  cfv 6561  t+ctcl 15024
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-trcl 15026
This theorem is referenced by:  rntrclfvRP  43744
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