Theorem predrelss 6276

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 5814	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆)
2		imass1 6039	. . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}))
3		sslin 4181	. . 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 6238	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		df-pred 6238	. 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋}))
7	4, 5, 6	3sstr4g 3977	1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   ∩ cin 3897   ⊆ wss 3898  {csn 4573  ^◡ccnv 5619   “ cima 5623  Predcpred 6237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238

This theorem is referenced by:  frmin  9606  frrlem16  9615  frr1  9616