Theorem 2ndval

Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	2ndval	$$⊢2^{nd}\left(A\right)=\bigcup \mathrm{ran}\left\{A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢x=A\to \left\{x\right\}=\left\{A\right\}$$
2	1	rneqd	$$⊢x=A\to \mathrm{ran}\left\{x\right\}=\mathrm{ran}\left\{A\right\}$$
3	2	unieqd	$$⊢x=A\to \bigcup \mathrm{ran}\left\{x\right\}=\bigcup \mathrm{ran}\left\{A\right\}$$
4		df-2nd	$$⊢2^{nd}=\left(x\in \mathrm{V}⟼\bigcup \mathrm{ran}\left\{x\right\}\right)$$
5		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
6	5	rnex	$$⊢\mathrm{ran}\left\{A\right\}\in \mathrm{V}$$
7	6	uniex	$$⊢\bigcup \mathrm{ran}\left\{A\right\}\in \mathrm{V}$$
8	3 4 7	fvmpt	$$⊢A\in \mathrm{V}\to 2^{nd}\left(A\right)=\bigcup \mathrm{ran}\left\{A\right\}$$
9		fvprc	$$⊢\neg A\in \mathrm{V}\to 2^{nd}\left(A\right)=\varnothing$$
10		snprc	$$⊢\neg A\in \mathrm{V}\leftrightarrow \left\{A\right\}=\varnothing$$
11	10	biimpi	$$⊢\neg A\in \mathrm{V}\to \left\{A\right\}=\varnothing$$
12	11	rneqd	$$⊢\neg A\in \mathrm{V}\to \mathrm{ran}\left\{A\right\}=\mathrm{ran}\varnothing$$
13		rn0	$$⊢\mathrm{ran}\varnothing =\varnothing$$
14	12 13	syl6eq	$$⊢\neg A\in \mathrm{V}\to \mathrm{ran}\left\{A\right\}=\varnothing$$
15	14	unieqd	$$⊢\neg A\in \mathrm{V}\to \bigcup \mathrm{ran}\left\{A\right\}=\bigcup \varnothing$$
16		uni0	$$⊢\bigcup \varnothing =\varnothing$$
17	15 16	syl6eq	$$⊢\neg A\in \mathrm{V}\to \bigcup \mathrm{ran}\left\{A\right\}=\varnothing$$
18	9 17	eqtr4d	$$⊢\neg A\in \mathrm{V}\to 2^{nd}\left(A\right)=\bigcup \mathrm{ran}\left\{A\right\}$$
19	8 18	pm2.61i	$$⊢2^{nd}\left(A\right)=\bigcup \mathrm{ran}\left\{A\right\}$$