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Mirrors > Home > MPE Home > Th. List > sess2 | Structured version Visualization version GIF version |
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
sess2 | ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssralv 3987 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V)) | |
2 | rabss2 4011 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥}) | |
3 | ssexg 5247 | . . . . . 6 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∧ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
4 | 3 | ex 413 | . . . . 5 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} → ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ({𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) |
6 | 5 | ralimdv 3109 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) |
7 | 1, 6 | syld 47 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)) |
8 | df-se 5545 | . 2 ⊢ (𝑅 Se 𝐵 ↔ ∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V) | |
9 | df-se 5545 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
10 | 7, 8, 9 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 Se wse 5542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rab 3073 df-v 3434 df-in 3894 df-ss 3904 df-se 5545 |
This theorem is referenced by: seeq2 5562 wereu2 5586 frpomin 6243 fprlem1 8116 wfrlem5OLD 8144 frmin 9507 frrlem15 9515 |
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