Theorem trclfv 14983

Description: The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)

Assertion

Ref	Expression
trclfv	⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Distinct variable group:   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclfv

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3480	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		trclexlem 14977	. . 3 ⊢ (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3		trclubg 14982	. . 3 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4	2, 3	ssexd 5325	. 2 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
5		trcleq1 14972	. . 3 ⊢ (𝑟 = 𝑅 → ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
6		df-trcl 14970	. . 3 ⊢ t+ = (𝑟 ∈ V ↦ ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
7	5, 6	fvmptg 7002	. 2 ⊢ ((𝑅 ∈ V ∧ ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
8	1, 4, 7	syl2anc2 583	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  Vcvv 3461   ∪ cun 3942   ⊆ wss 3944  ∩ cint 4950   × cxp 5676  dom cdm 5678  ran crn 5679   ∘ ccom 5682  ‘cfv 6549  t+ctcl 14968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-trcl 14970

This theorem is referenced by:  brtrclfv  14985  trclfvss  14989  trclfvub  14990  trclfvlb  14991  cotrtrclfv  14995  trclun  14997  brtrclfv2  43299