MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmtrclfv Structured version   Visualization version   GIF version

Theorem dmtrclfv 15054
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 15043 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 dmss 5916 . . . 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 5924 . . . 4 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5 dm0rn0 5938 . . . . . . 7 (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6 xpeq1 5703 . . . . . . . . . 10 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7 0xp 5787 . . . . . . . . . 10 (∅ × ran 𝑅) = ∅
86, 7eqtrdi 2791 . . . . . . . . 9 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
98dmeqd 5919 . . . . . . . 8 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10 dm0 5934 . . . . . . . . 9 dom ∅ = ∅
1110a1i 11 . . . . . . . 8 (dom 𝑅 = ∅ → dom ∅ = ∅)
12 eqcom 2742 . . . . . . . . 9 (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
1312biimpi 216 . . . . . . . 8 (dom 𝑅 = ∅ → ∅ = dom 𝑅)
149, 11, 133eqtrd 2779 . . . . . . 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 5942 . . . . . 6 (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
1715, 16pm2.61ine 3023 . . . . 5 dom (dom 𝑅 × ran 𝑅) = dom 𝑅
1817uneq2i 4175 . . . 4 (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19 unidm 4167 . . . 4 (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
204, 18, 193eqtri 2767 . . 3 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
213, 20sseqtrdi 4046 . 2 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22 trclfvlb 15044 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
23 dmss 5916 . . 3 (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
2422, 23syl 17 . 2 (𝑅𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
2521, 24eqssd 4013 1 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cun 3961  wss 3963  c0 4339   × cxp 5687  dom cdm 5689  ran crn 5690  cfv 6563  t+ctcl 15021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-trcl 15023
This theorem is referenced by:  rntrclfvRP  43721
  Copyright terms: Public domain W3C validator