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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 4228 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 6294 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 6294 | . 2 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
4 | 1, 2, 3 | 3sstr4g 4022 | 1 ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3942 ⊆ wss 3943 {csn 4623 ◡ccnv 5668 “ cima 5672 Predcpred 6293 |
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 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-pred 6294 |
This theorem is referenced by: preddowncl 6327 wfrlem8OLD 8317 |
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