Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  trclfvlb3 Structured version   Visualization version   GIF version

Theorem trclfvlb3 14411
 Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
Assertion
Ref Expression
trclfvlb3 (𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))

Proof of Theorem trclfvlb3
StepHypRef Expression
1 trclfvlb 14408 . 2 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
2 trclfvlb2 14410 . 2 (𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))
31, 2unssd 4092 1 (𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ∪ cun 3857   ⊆ wss 3859   ∘ ccom 5529  ‘cfv 6336  t+ctcl 14385 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6295  df-fun 6338  df-fv 6344  df-trcl 14387 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator