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Theorem trclfv 14360
Description: The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
Assertion
Ref Expression
trclfv (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Distinct variable group:   𝑥,𝑅
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclfv
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elex 3498 . 2 (𝑅𝑉𝑅 ∈ V)
2 trclexlem 14354 . . 3 (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3 trclubg 14359 . . 3 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
42, 3ssexd 5214 . 2 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
5 trcleq1 14349 . . 3 (𝑟 = 𝑅 {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
6 df-trcl 14347 . . 3 t+ = (𝑟 ∈ V ↦ {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
75, 6fvmptg 6757 . 2 ((𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
81, 4, 7syl2anc2 588 1 (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2802  Vcvv 3480  cun 3917  wss 3919   cint 4862   × cxp 5540  dom cdm 5542  ran crn 5543  ccom 5546  cfv 6343  t+ctcl 14345
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-trcl 14347
This theorem is referenced by:  brtrclfv  14362  trclfvss  14366  trclfvub  14367  trclfvlb  14368  cotrtrclfv  14372  trclun  14374  brtrclfv2  40344
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