Theorem predpredss 6308

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 4229	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (◡𝑅 “ {𝑋})))
2		df-pred 6301	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3		df-pred 6301	. 2 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
4	1, 2, 3	3sstr4g 4019	1 ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∩ cin 3940   ⊆ wss 3941  {csn 4625  ^◡ccnv 5672   “ cima 5676  Predcpred 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-in 3948  df-ss 3958  df-pred 6301

This theorem is referenced by:  preddowncl  6334  wfrlem8OLD  8330