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Metamath Proof Explorer

Theorem pwselbasb

Description: Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015)

Ref Expression
Hypotheses pwsbas.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
pwsbas.f ⊢ 𝐵 = ( Base ‘ 𝑅 )
pwselbas.v ⊢ 𝑉 = ( Base ‘ 𝑌 )
Assertion pwselbasb ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ 𝑉 ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 pwsbas.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
2 pwsbas.f ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 pwselbas.v ⊢ 𝑉 = ( Base ‘ 𝑌 )
4 1 2 pwsbas ⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝐵 ↑m 𝐼 ) = ( Base ‘ 𝑌 ) )
5 4 3 eqtr4di ⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝐵 ↑m 𝐼 ) = 𝑉 )
6 5 eleq2d ⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ↔ 𝑋 ∈ 𝑉 ) )
7 2 fvexi ⊢ 𝐵 ∈ V
8 elmapg ⊢ ( ( 𝐵 ∈ V ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )
9 7 8 mpan ⊢ ( 𝐼 ∈ 𝑍 → ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )
10 9 adantl ⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )
11 6 10 bitr3d ⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ 𝑉 ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )