Theorem predrelss 6255

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 5794	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆)
2		imass1 6019	. . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}))
3		sslin 4174	. . 3 ⊢ ((◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}) → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (◡𝑆 “ {𝑋})))
4	1, 2, 3	3syl 18	. 2 ⊢ (𝑅 ⊆ 𝑆 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐴 ∩ (◡𝑆 “ {𝑋})))
5		df-pred 6217	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		df-pred 6217	. 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋}))
7	4, 5, 6	3sstr4g 3971	1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   ∩ cin 3891   ⊆ wss 3892  {csn 4565  ^◡ccnv 5599   “ cima 5603  Predcpred 6216

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-cnv 5608  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217

This theorem is referenced by:  frmin  9555  frrlem16  9564  frr1  9565