Theorem fo1stres

Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008)

	Ref	Expression
Assertion	fo1stres	$$⊢B\ne \varnothing \to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}A$$

Proof

Step	Hyp	Ref	Expression
1		n0	$$⊢B\ne \varnothing \leftrightarrow \exists yy\in B$$
2		opelxp	$$⊢⟨x,y⟩\in \left(A\times B\right)\leftrightarrow \left(x\in A\wedge y\in B\right)$$
3		fvres	$$⊢⟨x,y⟩\in \left(A\times B\right)\to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)=1^{st}\left(⟨x,y⟩\right)$$
4		vex	$$⊢x\in \mathrm{V}$$
5		vex	$$⊢y\in \mathrm{V}$$
6	4 5	op1st	$$⊢1^{st}\left(⟨x,y⟩\right)=x$$
7	3 6	eqtr2di	$$⊢⟨x,y⟩\in \left(A\times B\right)\to x=\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)$$
8		f1stres	$$⊢\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶A$$
9		ffn	$$⊢\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶A\to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)$$
10	8 9	ax-mp	$$⊢\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)$$
11		fnfvelrn	$$⊢\left(\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)Fn\left(A\times B\right)\wedge ⟨x,y⟩\in \left(A\times B\right)\right)\to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)$$
12	10 11	mpan	$$⊢⟨x,y⟩\in \left(A\times B\right)\to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\left(⟨x,y⟩\right)\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)$$
13	7 12	eqeltrd	$$⊢⟨x,y⟩\in \left(A\times B\right)\to x\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)$$
14	2 13	sylbir	$$⊢\left(x\in A\wedge y\in B\right)\to x\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)$$
15	14	expcom	$$⊢y\in B\to \left(x\in A\to x\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
16	15	exlimiv	$$⊢\exists yy\in B\to \left(x\in A\to x\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
17	1 16	sylbi	$$⊢B\ne \varnothing \to \left(x\in A\to x\in \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
18	17	ssrdv	$$⊢B\ne \varnothing \to A\subseteq \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)$$
19		frn	$$⊢\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶A\to \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq A$$
20	8 19	ax-mp	$$⊢\mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq A$$
21	18 20	jctil	$$⊢B\ne \varnothing \to \left(\mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq A\wedge A\subseteq \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
22		eqss	$$⊢\mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)=A\leftrightarrow \left(\mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\subseteq A\wedge A\subseteq \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)\right)$$
23	21 22	sylibr	$$⊢B\ne \varnothing \to \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)=A$$
24	23 8	jctil	$$⊢B\ne \varnothing \to \left(\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶A\wedge \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)=A\right)$$
25		dffo2	$$⊢\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}A\leftrightarrow \left(\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B⟶A\wedge \mathrm{ran}\left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right)=A\right)$$
26	24 25	sylibr	$$⊢B\ne \varnothing \to \left({1^{st}\upharpoonright }_{\left(A\times B\right)}\right):A\times B\underset{onto}{⟶}A$$