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Theorem trclfvss 13791
 Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
Assertion
Ref Expression
trclfvss ((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆))

Proof of Theorem trclfvss
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 trclsslem 13775 . . 3 (𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
213ad2ant3 1104 . 2 ((𝑅𝑉𝑆𝑊𝑅𝑆) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3 trclfv 13785 . . 3 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
433ad2ant1 1102 . 2 ((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
5 trclfv 13785 . . 3 (𝑆𝑊 → (t+‘𝑆) = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
653ad2ant2 1103 . 2 ((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑆) = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
72, 4, 63sstr4d 3681 1 ((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637   ⊆ wss 3607  ∩ cint 4507   ∘ ccom 5147  ‘cfv 5926  t+ctcl 13770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-trcl 13772 This theorem is referenced by: (None)
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