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Mirrors > Home > MPE Home > Th. List > predrelss | Structured version Visualization version GIF version |
Description: Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.) |
Ref | Expression |
---|---|
predrelss | ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5885 | . . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆) | |
2 | imass1 6121 | . . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋})) | |
3 | sslin 4250 | . . 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 6322 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
6 | df-pred 6322 | . 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋})) | |
7 | 4, 5, 6 | 3sstr4g 4040 | 1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3961 ⊆ wss 3962 {csn 4630 ◡ccnv 5687 “ cima 5691 Predcpred 6321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 |
This theorem is referenced by: frmin 9786 frrlem16 9795 frr1 9796 |
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