Theorem trclfvub 15169

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

Assertion

Ref	Expression
trclfvub	⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Proof of Theorem trclfvub

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trclfv 15162	. 2 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
2		trclubg 15161	. 2 ⊢ (𝑅 ∈ 𝑉 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	eqsstrd 3943	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  {cab 2738   ⊆ wss 3896   ∪ cun 4148  ∩ cint 4906   × cxp 5649  dom cdm 5651  ran crn 5652   ∘ ccom 5655  ‘cfv 6543  t+ctcl 15147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-fv 6551  df-trcl 15149

This theorem is used by:  reltrclfv  15179  dmtrclfv  15180  rntrclfvOAI  43811