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Theorem trclfvlb 14361
 Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
Assertion
Ref Expression
trclfvlb (𝑅𝑉𝑅 ⊆ (t+‘𝑅))

Proof of Theorem trclfvlb
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 ssmin 4857 . 2 𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
2 trclfv 14353 . 2 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
31, 2sseqtrrid 3968 1 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {cab 2776   ⊆ wss 3881  ∩ cint 4838   ∘ ccom 5523  ‘cfv 6324  t+ctcl 14338 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-trcl 14340 This theorem is referenced by:  trclfvlb2  14363  trclfvlb3  14364  cotrtrclfv  14365  trclfvg  14368  dmtrclfv  14371  rntrclfvOAI  39647  brtrclfv2  40443  frege96d  40465  frege91d  40467  frege97d  40468  frege109d  40473  frege131d  40480
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