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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 1093 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓)) | |
2 | rabssrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) | |
3 | 1, 2 | sylbir 237 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
4 | 3 | ex 415 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
5 | 4 | ss2rabdv 4051 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
6 | rabssrabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | rabss2 4053 | . . 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 3976 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-in 3942 df-ss 3951 |
This theorem is referenced by: suppfnss 7849 clwlknon2num 28141 numclwlk1lem2 28143 |
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