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Theorem dmtrclfv 14366
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 14355 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 dmss 5764 . . . 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 5772 . . . 4 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5 dm0rn0 5788 . . . . . . 7 (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6 xpeq1 5562 . . . . . . . . . 10 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7 0xp 5642 . . . . . . . . . 10 (∅ × ran 𝑅) = ∅
86, 7syl6eq 2869 . . . . . . . . 9 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
98dmeqd 5767 . . . . . . . 8 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10 dm0 5783 . . . . . . . . 9 dom ∅ = ∅
1110a1i 11 . . . . . . . 8 (dom 𝑅 = ∅ → dom ∅ = ∅)
12 eqcom 2825 . . . . . . . . 9 (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
1312biimpi 217 . . . . . . . 8 (dom 𝑅 = ∅ → ∅ = dom 𝑅)
149, 11, 133eqtrd 2857 . . . . . . 7 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
155, 14sylbir 236 . . . . . 6 (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16 dmxp 5792 . . . . . 6 (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
1715, 16pm2.61ine 3097 . . . . 5 dom (dom 𝑅 × ran 𝑅) = dom 𝑅
1817uneq2i 4133 . . . 4 (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19 unidm 4125 . . . 4 (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
204, 18, 193eqtri 2845 . . 3 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
213, 20sseqtrdi 4014 . 2 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22 trclfvlb 14356 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
23 dmss 5764 . . 3 (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
2422, 23syl 17 . 2 (𝑅𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
2521, 24eqssd 3981 1 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cun 3931  wss 3933  c0 4288   × cxp 5546  dom cdm 5548  ran crn 5549  cfv 6348  t+ctcl 14333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-trcl 14335
This theorem is referenced by:  rntrclfvRP  39954
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