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Theorem trclfv 14973
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 3489 . 2 (𝑅𝑉𝑅 ∈ V)
2 trclexlem 14967 . . 3 (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3 trclubg 14972 . . 3 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
42, 3ssexd 5318 . 2 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
5 trcleq1 14962 . . 3 (𝑟 = 𝑅 {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
6 df-trcl 14960 . . 3 t+ = (𝑟 ∈ V ↦ {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
75, 6fvmptg 6997 . 2 ((𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
81, 4, 7syl2anc2 584 1 (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  Vcvv 3470  cun 3943  wss 3945   cint 4944   × cxp 5670  dom cdm 5672  ran crn 5673  ccom 5676  cfv 6542  t+ctcl 14958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-trcl 14960
This theorem is referenced by:  brtrclfv  14975  trclfvss  14979  trclfvub  14980  trclfvlb  14981  cotrtrclfv  14985  trclun  14987  brtrclfv2  43151
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