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Mirrors > Home > MPE Home > Th. List > rabss3d | Structured version Visualization version GIF version |
Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
Ref | Expression |
---|---|
rabss3d.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
rabss3d | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfrab1 3451 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓} | |
3 | nfrab1 3451 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓} | |
4 | rabss3d.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵) | |
5 | simprr 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓) | |
6 | 4, 5 | jca 512 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓)) |
7 | 6 | ex 413 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓))) |
8 | rabid 3452 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) | |
9 | rabid 3452 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) | |
10 | 7, 8, 9 | 3imtr4g 295 | . 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})) |
11 | 1, 2, 3, 10 | ssrd 3987 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 {crab 3432 ⊆ wss 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-rab 3433 df-v 3476 df-in 3955 df-ss 3965 |
This theorem is referenced by: xpinpreima2 33173 reprss 33915 reprinfz1 33920 upwrdfi 45900 |
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