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Theorem trclfvlb 14902
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 4932 . 2 𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
2 trclfv 14894 . 2 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
31, 2sseqtrrid 4001 1 (𝑅𝑉𝑅 ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  {cab 2710  wss 3914   cint 4911  ccom 5641  cfv 6500  t+ctcl 14879
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-trcl 14881
This theorem is referenced by:  trclfvlb2  14904  trclfvlb3  14905  cotrtrclfv  14906  trclfvg  14909  dmtrclfv  14912  rntrclfvOAI  41061  brtrclfv2  42091  frege96d  42113  frege91d  42115  frege97d  42116  frege109d  42121  frege131d  42128
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