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Theorem cotrtrclfv 14360
Description: The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
Assertion
Ref Expression
cotrtrclfv ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Proof of Theorem cotrtrclfv
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 trclfv 14348 . . . 4 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21adantr 481 . . 3 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3 simpr 485 . . . . . 6 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅𝑅) ⊆ 𝑅)
4 ssid 3986 . . . . . 6 𝑅𝑅
53, 4jctil 520 . . . . 5 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅))
6 trcleq2lem 14339 . . . . . . 7 (𝑟 = 𝑅 → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
76elabg 3663 . . . . . 6 (𝑅𝑉 → (𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
87adantr 481 . . . . 5 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
95, 8mpbird 258 . . . 4 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → 𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
10 intss1 4882 . . . 4 (𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ 𝑅)
119, 10syl 17 . . 3 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ 𝑅)
122, 11eqsstrd 4002 . 2 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) ⊆ 𝑅)
13 trclfvlb 14356 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
1413adantr 481 . 2 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → 𝑅 ⊆ (t+‘𝑅))
1512, 14eqssd 3981 1 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {cab 2796  wss 3933   cint 4867  ccom 5552  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:  trclidm  14361
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