Theorem fvdiagfn

Description: Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015)

	Ref	Expression
Hypothesis	fdiagfn.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( 𝐼 × { 𝑥 } ) )
Assertion	fvdiagfn	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐼 × { 𝑋 } ) )

Proof

Step	Hyp	Ref	Expression
1		fdiagfn.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( 𝐼 × { 𝑥 } ) )
2		sneq	⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } )
3	2	xpeq2d	⊢ ( 𝑥 = 𝑋 → ( 𝐼 × { 𝑥 } ) = ( 𝐼 × { 𝑋 } ) )
4		simpr	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
5		snex	⊢ { 𝑋 } ∈ V
6		xpexg	⊢ ( ( 𝐼 ∈ 𝑊 ∧ { 𝑋 } ∈ V ) → ( 𝐼 × { 𝑋 } ) ∈ V )
7	5 6	mpan2	⊢ ( 𝐼 ∈ 𝑊 → ( 𝐼 × { 𝑋 } ) ∈ V )
8	7	adantr	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 × { 𝑋 } ) ∈ V )
9	1 3 4 8	fvmptd3	⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐼 × { 𝑋 } ) )