Theorem xpsfrn

Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015)

	Ref	Expression
Hypothesis	xpsff1o.f	$$⊢F=\left(x\in A,y\in B⟼\left\{⟨\varnothing ,x⟩,⟨1_{𝑜},y⟩\right\}\right)$$
Assertion	xpsfrn	$$⊢\mathrm{ran}F=\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$

Proof

Step	Hyp	Ref	Expression
1		xpsff1o.f	$$⊢F=\left(x\in A,y\in B⟼\left\{⟨\varnothing ,x⟩,⟨1_{𝑜},y⟩\right\}\right)$$
2	1	xpsff1o	$$⊢F:A\times B\underset{1-1\; onto}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$
3		f1ofo	$$⊢F:A\times B\underset{1-1\; onto}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)\to F:A\times B\underset{onto}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$
4		forn	$$⊢F:A\times B\underset{onto}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)\to \mathrm{ran}F=\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$
5	2 3 4	mp2b	$$⊢\mathrm{ran}F=\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$