Theorem trclfv 15049

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 3509	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		trclexlem 15043	. . 3 ⊢ (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3		trclubg 15048	. . 3 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4	2, 3	ssexd 5342	. 2 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
5		trcleq1 15038	. . 3 ⊢ (𝑟 = 𝑅 → ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
6		df-trcl 15036	. . 3 ⊢ t+ = (𝑟 ∈ V ↦ ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
7	5, 6	fvmptg 7027	. 2 ⊢ ((𝑅 ∈ V ∧ ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
8	1, 4, 7	syl2anc2 584	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ∩ cint 4970   × cxp 5698  dom cdm 5700  ran crn 5701   ∘ ccom 5704  ‘cfv 6573  t+ctcl 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-trcl 15036

This theorem is referenced by:  brtrclfv  15051  trclfvss  15055  trclfvub  15056  trclfvlb  15057  cotrtrclfv  15061  trclun  15063  brtrclfv2  43689