Theorem predpredss 6274

Description: If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predpredss	⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predpredss

Step	Hyp	Ref	Expression
1		ssrin 4196	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6267	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6267	. 2 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
4	1, 2, 3	3sstr4g 3989	1 ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∩ cin 3902   ⊆ wss 3903  {csn 4582  ^◡ccnv 5631   “ cima 5635  Predcpred 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ss 3920  df-pred 6267

This theorem is referenced by:  preddowncl  6298