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| Mirrors > Home > MPE Home > Th. List > rabssrabd | Structured version Visualization version GIF version | ||
| Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.) |
| Ref | Expression |
|---|---|
| rabssrabd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| rabssrabd.2 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) |
| Ref | Expression |
|---|---|
| rabssrabd | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3anan32 1097 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓)) | |
| 2 | rabssrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) | |
| 3 | 1, 2 | sylbir 235 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
| 4 | 3 | ex 412 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
| 5 | 4 | ss2rabdv 4076 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
| 6 | rabssrabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 7 | rabss2 4078 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) | |
| 8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
| 9 | 5, 8 | sstrd 3994 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 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-ss 3968 |
| This theorem is referenced by: suppfnss 8214 clwlknon2num 30387 numclwlk1lem2 30389 |
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