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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 1912 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfrab1 3454 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓} | |
3 | nfrab1 3454 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐵 ∣ 𝜓} | |
4 | rabss3d.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝑥 ∈ 𝐵) | |
5 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜓) | |
6 | 4, 5 | jca 511 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → (𝑥 ∈ 𝐵 ∧ 𝜓)) |
7 | 6 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜓))) |
8 | rabid 3455 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) | |
9 | rabid 3455 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) | |
10 | 7, 8, 9 | 3imtr4g 296 | . 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} → 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})) |
11 | 1, 2, 3, 10 | ssrd 4000 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-ss 3980 |
This theorem is referenced by: xpinpreima2 33868 reprss 34611 reprinfz1 34616 unitscyglem2 42178 unitscyglem4 42180 unitscyglem5 42181 upwrdfi 46841 |
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