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| Mirrors > Home > MPE Home > Th. List > ressbas2 | Structured version Visualization version GIF version | ||
| Description: Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| Ref | Expression |
|---|---|
| ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
| ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
| Ref | Expression |
|---|---|
| ressbas2 | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfss2 3969 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
| 3 | ressbas.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
| 4 | 3 | fvexi 6920 | . . . 4 ⊢ 𝐵 ∈ V |
| 5 | 4 | ssex 5321 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
| 6 | ressbas.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
| 7 | 6, 3 | ressbas 17280 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
| 8 | 5, 7 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
| 9 | 2, 8 | eqtr3d 2779 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑅)) |
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