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Mirrors > Home > MPE Home > Th. List > fvmptd3 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 6529. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fvmptd3.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
fvmptd3.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptd3.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd3.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmptd3 | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd3.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | fvmptd3.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
3 | nfcv 2969 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2969 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
5 | fvmptd3.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
6 | fvmptd3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
7 | 3, 4, 5, 6 | fvmptf 6548 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
8 | 1, 2, 7 | syl2anc 581 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
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