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| Mirrors > Home > MPE Home > Th. List > fvmptd | Structured version Visualization version GIF version | ||
| Description: Deduction version of fvmpt 7016. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.) |
| Ref | Expression |
|---|---|
| fvmptd.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
| fvmptd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
| fvmptd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
| fvmptd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| fvmptd | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvmptd.1 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
| 2 | fvmptd.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
| 3 | fvmptd.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
| 4 | fvmptd.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
| 5 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 6 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 7 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | fvmptdf 7022 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
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