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Theorem predrelss 6359
Description: Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)
Assertion
Ref Expression
predrelss (𝑅𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

Proof of Theorem predrelss
StepHypRef Expression
1 cnvss 5885 . . 3 (𝑅𝑆𝑅𝑆)
2 imass1 6121 . . 3 (𝑅𝑆 → (𝑅 “ {𝑋}) ⊆ (𝑆 “ {𝑋}))
3 sslin 4250 . . 3 ((𝑅 “ {𝑋}) ⊆ (𝑆 “ {𝑋}) → (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (𝑆 “ {𝑋})))
41, 2, 33syl 18 . 2 (𝑅𝑆 → (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (𝑆 “ {𝑋})))
5 df-pred 6322 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
6 df-pred 6322 . 2 Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (𝑆 “ {𝑋}))
74, 5, 63sstr4g 4040 1 (𝑅𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3961  wss 3962  {csn 4630  ccnv 5687  cima 5691  Predcpred 6321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322
This theorem is referenced by:  frmin  9786  frrlem16  9795  frr1  9796
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