Metamath Proof Explorer

Theorem fvixp

Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

Ref Expression
Hypothesis fvixp.1 ⊢ x = C → B = D
Assertion fvixp ⊢ F ∈ ⨉ x ∈ A B ∧ C ∈ A → F ⁡ C ∈ D

Proof

Step Hyp Ref Expression
1 fvixp.1 ⊢ x = C → B = D
2 elixp2 ⊢ F ∈ ⨉ x ∈ A B ↔ F ∈ V ∧ F Fn A ∧ ∀ x ∈ A F ⁡ x ∈ B
3 2 simp3bi ⊢ F ∈ ⨉ x ∈ A B → ∀ x ∈ A F ⁡ x ∈ B
4 fveq2 ⊢ x = C → F ⁡ x = F ⁡ C
5 4 1 eleq12d ⊢ x = C → F ⁡ x ∈ B ↔ F ⁡ C ∈ D
6 5 rspccva ⊢ ∀ x ∈ A F ⁡ x ∈ B ∧ C ∈ A → F ⁡ C ∈ D
7 3 6 sylan ⊢ F ∈ ⨉ x ∈ A B ∧ C ∈ A → F ⁡ C ∈ D