Theorem predrelss 6345

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 5863	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆)
2		imass1 6108	. . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋}))
3		sslin 4198	. . 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 6309	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		df-pred 6309	. 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋}))
7	4, 5, 6	3sstr4g 3993	1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3907   ⊆ wss 3908  {csn 4594  ^◡ccnv 5665   “ cima 5669  Predcpred 6308

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 2148  ax-9 2156  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309

This theorem is used by:  frmin  9731  frrlem16  9740  frr1  9741