Theorem trclfv 14711

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 3450	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		trclexlem 14705	. . 3 ⊢ (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3		trclubg 14710	. . 3 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4	2, 3	ssexd 5248	. 2 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
5		trcleq1 14700	. . 3 ⊢ (𝑟 = 𝑅 → ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
6		df-trcl 14698	. . 3 ⊢ t+ = (𝑟 ∈ V ↦ ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
7	5, 6	fvmptg 6873	. 2 ⊢ ((𝑅 ∈ V ∧ ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
8	1, 4, 7	syl2anc2 585	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∩ cint 4879   × cxp 5587  dom cdm 5589  ran crn 5590   ∘ ccom 5593  ‘cfv 6433  t+ctcl 14696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-trcl 14698

This theorem is referenced by:  brtrclfv  14713  trclfvss  14717  trclfvub  14718  trclfvlb  14719  cotrtrclfv  14723  trclun  14725  brtrclfv2  41335