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Theorem dmtrclfv 14369
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 14358 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 dmss 5748 . . . 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 5756 . . . 4 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5 dm0rn0 5772 . . . . . . 7 (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6 xpeq1 5546 . . . . . . . . . 10 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7 0xp 5626 . . . . . . . . . 10 (∅ × ran 𝑅) = ∅
86, 7syl6eq 2873 . . . . . . . . 9 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
98dmeqd 5751 . . . . . . . 8 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10 dm0 5767 . . . . . . . . 9 dom ∅ = ∅
1110a1i 11 . . . . . . . 8 (dom 𝑅 = ∅ → dom ∅ = ∅)
12 eqcom 2829 . . . . . . . . 9 (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
1312biimpi 219 . . . . . . . 8 (dom 𝑅 = ∅ → ∅ = dom 𝑅)
149, 11, 133eqtrd 2861 . . . . . . 7 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
155, 14sylbir 238 . . . . . 6 (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16 dmxp 5776 . . . . . 6 (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
1715, 16pm2.61ine 3094 . . . . 5 dom (dom 𝑅 × ran 𝑅) = dom 𝑅
1817uneq2i 4111 . . . 4 (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19 unidm 4103 . . . 4 (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
204, 18, 193eqtri 2849 . . 3 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
213, 20sseqtrdi 3992 . 2 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22 trclfvlb 14359 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
23 dmss 5748 . . 3 (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
2422, 23syl 17 . 2 (𝑅𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
2521, 24eqssd 3959 1 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  cun 3906  wss 3908  c0 4265   × cxp 5530  dom cdm 5532  ran crn 5533  cfv 6334  t+ctcl 14336
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-trcl 14338
This theorem is referenced by:  rntrclfvRP  40362
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