Theorem ackbij1lem7

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

	Ref	Expression
Hypothesis	ackbij.f	$$⊢F=\left(x\in \left(𝒫\mathrm{\omega }\cap \mathrm{Fin}\right)⟼\mathrm{card}\left(\bigcup _{y\in x}\left(\left\{y\right\}\times 𝒫y\right)\right)\right)$$
Assertion	ackbij1lem7	$$⊢A\in \left(𝒫\mathrm{\omega }\cap \mathrm{Fin}\right)\to F\left(A\right)=\mathrm{card}\left(\bigcup _{y\in A}\left(\left\{y\right\}\times 𝒫y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ackbij.f	$$⊢F=\left(x\in \left(𝒫\mathrm{\omega }\cap \mathrm{Fin}\right)⟼\mathrm{card}\left(\bigcup _{y\in x}\left(\left\{y\right\}\times 𝒫y\right)\right)\right)$$
2		iuneq1	$$⊢x=A\to \bigcup _{y\in x}\left(\left\{y\right\}\times 𝒫y\right)=\bigcup _{y\in A}\left(\left\{y\right\}\times 𝒫y\right)$$
3	2	fveq2d	$$⊢x=A\to \mathrm{card}\left(\bigcup _{y\in x}\left(\left\{y\right\}\times 𝒫y\right)\right)=\mathrm{card}\left(\bigcup _{y\in A}\left(\left\{y\right\}\times 𝒫y\right)\right)$$
4		fvex	$$⊢\mathrm{card}\left(\bigcup _{y\in A}\left(\left\{y\right\}\times 𝒫y\right)\right)\in \mathrm{V}$$
5	3 1 4	fvmpt	$$⊢A\in \left(𝒫\mathrm{\omega }\cap \mathrm{Fin}\right)\to F\left(A\right)=\mathrm{card}\left(\bigcup _{y\in A}\left(\left\{y\right\}\times 𝒫y\right)\right)$$