Theorem trclfvlb2 14127

Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)

Assertion

Ref	Expression
trclfvlb2	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ 𝑅) ⊆ (t+‘𝑅))

Proof of Theorem trclfvlb2

Step	Hyp	Ref	Expression
1		trclfvcotr 14126	. 2 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2		trclfvlb 14125	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
3	1, 2, 2	trrelssd 14090	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ 𝑅) ⊆ (t+‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2166   ⊆ wss 3797   ∘ ccom 5345  ‘cfv 6122  t+ctcl 14102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-int 4697  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-iota 6085  df-fun 6124  df-fv 6130  df-trcl 14104

This theorem is referenced by:  trclfvlb3  14128