Theorem trclfv 14363

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 3515	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		trclexlem 14357	. . 3 ⊢ (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3		trclubg 14362	. . 3 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4	2, 3	ssexd 5231	. 2 ⊢ (𝑅 ∈ V → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
5		trcleq1 14352	. . 3 ⊢ (𝑟 = 𝑅 → ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
6		df-trcl 14350	. . 3 ⊢ t+ = (𝑟 ∈ V ↦ ∩ {𝑥 ∣ (𝑟 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
7	5, 6	fvmptg 6769	. 2 ⊢ ((𝑅 ∈ V ∧ ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
8	1, 4, 7	syl2anc2 587	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {cab 2802  Vcvv 3497   ∪ cun 3937   ⊆ wss 3939  ∩ cint 4879   × cxp 5556  dom cdm 5558  ran crn 5559   ∘ ccom 5562  ‘cfv 6358  t+ctcl 14348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fv 6366  df-trcl 14350

This theorem is referenced by:  brtrclfv  14365  trclfvss  14369  trclfvub  14370  trclfvlb  14371  cotrtrclfv  14375  trclun  14377  brtrclfv2  40078