Theorem xpsff1o2

Description: The function appearing in xpsval is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = { (/) , 1o } . (Contributed by Mario Carneiro, 24-Jan-2015)

	Ref	Expression
Hypothesis	xpsff1o.f	$$⊢F=\left(x\in A,y\in B⟼\left\{⟨\varnothing ,x⟩,⟨1_{𝑜},y⟩\right\}\right)$$
Assertion	xpsff1o2	$$⊢F:A\times B\underset{1-1\; onto}{⟶}\mathrm{ran}F$$

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		f1of1	$$⊢F:A\times B\underset{1-1\; onto}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)\to F:A\times B\underset{1-1}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)$$
4		f1f1orn	$$⊢F:A\times B\underset{1-1}{⟶}\underset{k\in 2_{𝑜}}{⨉}if\left(k=\varnothing ,A,B\right)\to F:A\times B\underset{1-1\; onto}{⟶}\mathrm{ran}F$$
5	2 3 4	mp2b	$$⊢F:A\times B\underset{1-1\; onto}{⟶}\mathrm{ran}F$$