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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 5883 | . . 3 ⊢ (𝑅 ⊆ 𝑆 → ◡𝑅 ⊆ ◡𝑆) | |
| 2 | imass1 6119 | . . 3 ⊢ (◡𝑅 ⊆ ◡𝑆 → (◡𝑅 “ {𝑋}) ⊆ (◡𝑆 “ {𝑋})) | |
| 3 | sslin 4243 | . . 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 6321 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
| 6 | df-pred 6321 | . 2 ⊢ Pred(𝑆, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑆 “ {𝑋})) | |
| 7 | 4, 5, 6 | 3sstr4g 4037 | 1 ⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ◡ccnv 5684 “ cima 5688 Predcpred 6320 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 |
| This theorem is referenced by: frmin 9789 frrlem16 9798 frr1 9799 |
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