Theorem predrelss 6339

Description: Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)

Assertion

Ref	Expression
predrelss	⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

Proof of Theorem predrelss

Step	Hyp	Ref	Expression
1		cnvss 5856	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆)
2		imass1 6101	. . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}))
3		sslin 4191	. . 3 ⊢ ((◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}) → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (◡𝑆 “ {𝑋})))
4	1, 2, 3	3syl 19	. 2 ⊢ (𝑅 ⊆ 𝑆 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (◡𝑆 “ {𝑋})))
5		df-pred 6303	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		df-pred 6303	. 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋}))
7	4, 5, 6	3sstr4g 3987	1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3901   ⊆ wss 3902  {csn 4587  ^◡ccnv 5658   “ cima 5662  Predcpred 6302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303

This theorem is used by:  frmin  9735  frrlem16  9744  frr1  9745