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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 1099 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓)) | |
2 | rabssrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) | |
3 | 1, 2 | sylbir 238 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
4 | 3 | ex 416 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
5 | 4 | ss2rabdv 3979 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
6 | rabssrabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | rabss2 3981 | . . 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 3901 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 {crab 3058 ⊆ wss 3857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rab 3063 df-v 3403 df-in 3864 df-ss 3874 |
This theorem is referenced by: suppfnss 7920 clwlknon2num 28423 numclwlk1lem2 28425 |
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