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Mirrors > Home > MPE Home > Th. List > predpredss | Structured version Visualization version GIF version |
Description: If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
predpredss | ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 3800 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 5597 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 5597 | . 2 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
4 | 1, 2, 3 | 3sstr4g 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3539 ⊆ wss 3540 {csn 4125 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-pred 5597 |
This theorem is referenced by: preddowncl 5624 wfrlem8 7309 |
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