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Metamath Proof Explorer

Theorem fmptf

Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fmptf.1 ⊢ Ⅎ 𝑥 𝐵
fmptf.2 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )
Assertion fmptf ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 )

Proof

Step Hyp Ref Expression
1 fmptf.1 ⊢ Ⅎ 𝑥 𝐵
2 fmptf.2 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 )
3 nfv ⊢ Ⅎ 𝑦 𝐶 ∈ 𝐵
4 nfcsb1v ⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐶
5 4 1 nfel ⊢ Ⅎ 𝑥 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ 𝐵
6 csbeq1a ⊢ ( 𝑥 = 𝑦 → 𝐶 = ⦋ 𝑦 / 𝑥 ⦌ 𝐶 )
7 6 eleq1d ⊢ ( 𝑥 = 𝑦 → ( 𝐶 ∈ 𝐵 ↔ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ 𝐵 ) )
8 3 5 7 cbvralw ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ 𝐵 )
9 nfcv ⊢ Ⅎ 𝑦 𝐶
10 9 4 6 cbvmpt ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐴 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 )
11 2 10 eqtri ⊢ 𝐹 = ( 𝑦 ∈ 𝐴 ↦ ⦋ 𝑦 / 𝑥 ⦌ 𝐶 )
12 11 fmpt ⊢ ( ∀ 𝑦 ∈ 𝐴 ⦋ 𝑦 / 𝑥 ⦌ 𝐶 ∈ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 )
13 8 12 bitri ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 )