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

Theorem dmtrclfv 13553
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 13542 . . . 4 (𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2 dmss 5232 . . . 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 5240 . . . 4 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅))
5 dm0rn0 5250 . . . . . . 7 (dom 𝑅 = ∅ ↔ ran 𝑅 = ∅)
6 xpeq1 5042 . . . . . . . . . 10 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = (∅ × ran 𝑅))
7 0xp 5112 . . . . . . . . . 10 (∅ × ran 𝑅) = ∅
86, 7syl6eq 2659 . . . . . . . . 9 (dom 𝑅 = ∅ → (dom 𝑅 × ran 𝑅) = ∅)
98dmeqd 5235 . . . . . . . 8 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom ∅)
10 dm0 5247 . . . . . . . . 9 dom ∅ = ∅
1110a1i 11 . . . . . . . 8 (dom 𝑅 = ∅ → dom ∅ = ∅)
12 eqcom 2616 . . . . . . . . 9 (dom 𝑅 = ∅ ↔ ∅ = dom 𝑅)
1312biimpi 204 . . . . . . . 8 (dom 𝑅 = ∅ → ∅ = dom 𝑅)
149, 11, 133eqtrd 2647 . . . . . . 7 (dom 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
155, 14sylbir 223 . . . . . 6 (ran 𝑅 = ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
16 dmxp 5252 . . . . . 6 (ran 𝑅 ≠ ∅ → dom (dom 𝑅 × ran 𝑅) = dom 𝑅)
1715, 16pm2.61ine 2864 . . . . 5 dom (dom 𝑅 × ran 𝑅) = dom 𝑅
1817uneq2i 3725 . . . 4 (dom 𝑅 ∪ dom (dom 𝑅 × ran 𝑅)) = (dom 𝑅 ∪ dom 𝑅)
19 unidm 3717 . . . 4 (dom 𝑅 ∪ dom 𝑅) = dom 𝑅
204, 18, 193eqtri 2635 . . 3 dom (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = dom 𝑅
213, 20syl6sseq 3613 . 2 (𝑅𝑉 → dom (t+‘𝑅) ⊆ dom 𝑅)
22 trclfvlb 13543 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
23 dmss 5232 . . 3 (𝑅 ⊆ (t+‘𝑅) → dom 𝑅 ⊆ dom (t+‘𝑅))
2422, 23syl 17 . 2 (𝑅𝑉 → dom 𝑅 ⊆ dom (t+‘𝑅))
2521, 24eqssd 3584 1 (𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cun 3537  wss 3539  c0 3873   × cxp 5026  dom cdm 5028  ran crn 5029  cfv 5790  t+ctcl 13518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798  df-trcl 13520
This theorem is referenced by:  rntrclfvRP  36838
  Copyright terms: Public domain W3C validator