Theorem ackbij1lem7

Description: Lemma for ackbij1 . (Contributed by Stefan O'Rear, 21-Nov-2014)

	Ref	Expression
Hypothesis	ackbij.f	⊢ 𝐹 = ( 𝑥 ∈ ( 𝒫 ω ∩ Fin ) ↦ ( card ‘ ∪ 𝑦 ∈ 𝑥 ( { 𝑦 } × 𝒫 𝑦 ) ) )
Assertion	ackbij1lem7	⊢ ( 𝐴 ∈ ( 𝒫 ω ∩ Fin ) → ( 𝐹 ‘ 𝐴 ) = ( card ‘ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝒫 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ackbij.f	⊢ 𝐹 = ( 𝑥 ∈ ( 𝒫 ω ∩ Fin ) ↦ ( card ‘ ∪ 𝑦 ∈ 𝑥 ( { 𝑦 } × 𝒫 𝑦 ) ) )
2		iuneq1	⊢ ( 𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ( { 𝑦 } × 𝒫 𝑦 ) = ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝒫 𝑦 ) )
3	2	fveq2d	⊢ ( 𝑥 = 𝐴 → ( card ‘ ∪ 𝑦 ∈ 𝑥 ( { 𝑦 } × 𝒫 𝑦 ) ) = ( card ‘ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝒫 𝑦 ) ) )
4		fvex	⊢ ( card ‘ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝒫 𝑦 ) ) ∈ V
5	3 1 4	fvmpt	⊢ ( 𝐴 ∈ ( 𝒫 ω ∩ Fin ) → ( 𝐹 ‘ 𝐴 ) = ( card ‘ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝒫 𝑦 ) ) )