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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 1089 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓)) | |
2 | rabssrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) | |
3 | 1, 2 | sylbir 236 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
4 | 3 | ex 413 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
5 | 4 | ss2rabdv 4049 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
6 | rabssrabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | rabss2 4051 | . . 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 3974 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-in 3940 df-ss 3949 |
This theorem is referenced by: suppfnss 7844 clwlknon2num 28074 numclwlk1lem2 28076 |
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