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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 4160 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ (𝐵 ∩ (◡𝑅 “ {𝑋}))) | |
2 | df-pred 6116 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
3 | df-pred 6116 | . 2 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋})) | |
4 | 1, 2, 3 | 3sstr4g 3960 | 1 ⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3880 ⊆ wss 3881 {csn 4525 ◡ccnv 5518 “ cima 5522 Predcpred 6115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-pred 6116 |
This theorem is referenced by: preddowncl 6143 wfrlem8 7945 |
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