Theorem fo2ndres

Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008)

	Ref	Expression
Assertion	fo2ndres	$$⊢A\ne \varnothing \to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}B$$

Proof

Step	Hyp	Ref	Expression
1		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
2		opelxp	$$⊢⟨x,y⟩\in \left(A\times B\right)\leftrightarrow \left(x\in A\wedge y\in B\right)$$
3		fvres	$$⊢⟨x,y⟩\in \left(A\times B\right)\to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)=2^{nd}\left(⟨x,y⟩\right)$$
4		vex	$$⊢x\in \mathrm{V}$$
5		vex	$$⊢y\in \mathrm{V}$$
6	4 5	op2nd	$$⊢2^{nd}\left(⟨x,y⟩\right)=y$$
7	3 6	eqtr2di	$$⊢⟨x,y⟩\in \left(A\times B\right)\to y=\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)$$
8		f2ndres	$$⊢\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶B$$
9		ffn	$$⊢\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶B\to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)$$
10	8 9	ax-mp	$$⊢\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)$$
11		fnfvelrn	$$⊢\left(\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)\wedge ⟨x,y⟩\in \left(A\times B\right)\right)\to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)$$
12	10 11	mpan	$$⊢⟨x,y⟩\in \left(A\times B\right)\to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)$$
13	7 12	eqeltrd	$$⊢⟨x,y⟩\in \left(A\times B\right)\to y\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)$$
14	2 13	sylbir	$$⊢\left(x\in A\wedge y\in B\right)\to y\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)$$
15	14	ex	$$⊢x\in A\to \left(y\in B\to y\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
16	15	exlimiv	$$⊢\exists xx\in A\to \left(y\in B\to y\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
17	1 16	sylbi	$$⊢A\ne \varnothing \to \left(y\in B\to y\in \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
18	17	ssrdv	$$⊢A\ne \varnothing \to B\subseteq \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)$$
19		frn	$$⊢\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶B\to \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq B$$
20	8 19	ax-mp	$$⊢\mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq B$$
21	18 20	jctil	$$⊢A\ne \varnothing \to \left(\mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq B\wedge B\subseteq \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
22		eqss	$$⊢\mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)=B\leftrightarrow \left(\mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq B\wedge B\subseteq \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
23	21 22	sylibr	$$⊢A\ne \varnothing \to \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)=B$$
24	23 8	jctil	$$⊢A\ne \varnothing \to \left(\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶B\wedge \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)=B\right)$$
25		dffo2	$$⊢\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}B\leftrightarrow \left(\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶B\wedge \mathrm{ran}\left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right)=B\right)$$
26	24 25	sylibr	$$⊢A\ne \varnothing \to \left({2^{nd}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}B$$