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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 1096 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓)) | |
2 | rabssrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜒) | |
3 | 1, 2 | sylbir 234 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
4 | 3 | ex 413 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
5 | 4 | ss2rabdv 4014 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
6 | rabssrabd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | rabss2 4016 | . . 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 3936 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rab 3075 df-v 3433 df-in 3899 df-ss 3909 |
This theorem is referenced by: suppfnss 7996 clwlknon2num 28728 numclwlk1lem2 28730 |
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