|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sess1 | Structured version Visualization version GIF version | ||
| Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| sess1 | ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . . . . . 6 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → 𝑅 ⊆ 𝑆) | |
| 2 | 1 | ssbrd 5186 | . . . . 5 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → 𝑦𝑆𝑥)) | 
| 3 | 2 | ss2rabdv 4076 | . . . 4 ⊢ (𝑅 ⊆ 𝑆 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥}) | 
| 4 | ssexg 5323 | . . . . 5 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
| 5 | 4 | ex 412 | . . . 4 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) | 
| 6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑅 ⊆ 𝑆 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) | 
| 7 | 6 | ralimdv 3169 | . 2 ⊢ (𝑅 ⊆ 𝑆 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) | 
| 8 | df-se 5638 | . 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) | |
| 9 | df-se 5638 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
| 10 | 7, 8, 9 | 3imtr4g 296 | 1 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 Se wse 5635 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-br 5144 df-se 5638 | 
| This theorem is referenced by: seeq1 5655 | 
| Copyright terms: Public domain | W3C validator |