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Theorem predrelss 6358
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 5883 . . 3 (𝑅𝑆𝑅𝑆)
2 imass1 6119 . . 3 (𝑅𝑆 → (𝑅 “ {𝑋}) ⊆ (𝑆 “ {𝑋}))
3 sslin 4243 . . 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 6321 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
6 df-pred 6321 . 2 Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (𝑆 “ {𝑋}))
74, 5, 63sstr4g 4037 1 (𝑅𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3950  wss 3951  {csn 4626  ccnv 5684  cima 5688  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  frmin  9789  frrlem16  9798  frr1  9799
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