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Theorem cotrtrclfv 14363
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 14351 . . . 4 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21adantr 484 . . 3 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3 simpr 488 . . . . . 6 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅𝑅) ⊆ 𝑅)
4 ssid 3964 . . . . . 6 𝑅𝑅
53, 4jctil 523 . . . . 5 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅))
6 trcleq2lem 14342 . . . . . . 7 (𝑟 = 𝑅 → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
76elabg 3641 . . . . . 6 (𝑅𝑉 → (𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
87adantr 484 . . . . 5 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
95, 8mpbird 260 . . . 4 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → 𝑅 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
10 intss1 4866 . . . 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 3980 . 2 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) ⊆ 𝑅)
13 trclfvlb 14359 . . 3 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
1413adantr 484 . 2 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → 𝑅 ⊆ (t+‘𝑅))
1512, 14eqssd 3959 1 ((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  {cab 2800  wss 3908   cint 4851  ccom 5536  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:  trclidm  14364
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