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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 486 | . . . . . 6 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → 𝑅 ⊆ 𝑆) | |
2 | 1 | ssbrd 5101 | . . . . 5 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → 𝑦𝑆𝑥)) |
3 | 2 | ss2rabdv 3994 | . . . 4 ⊢ (𝑅 ⊆ 𝑆 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥}) |
4 | ssexg 5221 | . . . . 5 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
5 | 4 | ex 416 | . . . 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 3101 | . 2 ⊢ (𝑅 ⊆ 𝑆 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) |
8 | df-se 5515 | . 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) | |
9 | df-se 5515 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
10 | 7, 8, 9 | 3imtr4g 299 | 1 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 {crab 3065 Vcvv 3413 ⊆ wss 3871 class class class wbr 5058 Se wse 5512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 df-v 3415 df-in 3878 df-ss 3888 df-br 5059 df-se 5515 |
This theorem is referenced by: seeq1 5528 |
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