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Theorem trclfv 14148
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 3413 . . 3 (𝑅𝑉𝑅 ∈ V)
2 trclexlem 14142 . . . 4 (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3 trclubg 14147 . . . 4 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
42, 3ssexd 5042 . . 3 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
51, 4jccir 517 . 2 (𝑅𝑉 → (𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V))
6 trcleq1 14137 . . 3 (𝑟 = 𝑅 {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
7 df-trcl 14135 . . 3 t+ = (𝑟 ∈ V ↦ {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
86, 7fvmptg 6540 . 2 ((𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
95, 8syl 17 1 (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  {cab 2762  Vcvv 3397  cun 3789  wss 3791   cint 4710   × cxp 5353  dom cdm 5355  ran crn 5356  ccom 5359  cfv 6135  t+ctcl 14133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-trcl 14135
This theorem is referenced by:  brtrclfv  14150  trclfvss  14154  trclfvub  14155  trclfvlb  14156  cotrtrclfv  14160  trclun  14162  brtrclfv2  38969
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