Theorem 2arymaptf1o

Description: The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations. (Contributed by AV, 23-May-2024)

	Ref	Expression
Hypothesis	2arymaptf.h	$$⊢H=\left(h\in \left(2\mathrm{-aryF}X\right)⟼\left(x\in X,y\in X⟼h\left(\left\{⟨0,x⟩,⟨1,y⟩\right\}\right)\right)\right)$$
Assertion	2arymaptf1o	$$⊢X\in V\to H:2\mathrm{-aryF}X\underset{1-1\; onto}{⟶}{X}^{\left(X\times X\right)}$$

Proof

Step	Hyp	Ref	Expression
1		2arymaptf.h	$$⊢H=\left(h\in \left(2\mathrm{-aryF}X\right)⟼\left(x\in X,y\in X⟼h\left(\left\{⟨0,x⟩,⟨1,y⟩\right\}\right)\right)\right)$$
2	1	2arymaptf1	$$⊢X\in V\to H:2\mathrm{-aryF}X\underset{1-1}{⟶}{X}^{\left(X\times X\right)}$$
3	1	2arymaptfo	$$⊢X\in V\to H:2\mathrm{-aryF}X\underset{onto}{⟶}{X}^{\left(X\times X\right)}$$
4		df-f1o	$$⊢H:2\mathrm{-aryF}X\underset{1-1\; onto}{⟶}{X}^{\left(X\times X\right)}\leftrightarrow \left(H:2\mathrm{-aryF}X\underset{1-1}{⟶}{X}^{\left(X\times X\right)}\wedge H:2\mathrm{-aryF}X\underset{onto}{⟶}{X}^{\left(X\times X\right)}\right)$$
5	2 3 4	sylanbrc	$$⊢X\in V\to H:2\mathrm{-aryF}X\underset{1-1\; onto}{⟶}{X}^{\left(X\times X\right)}$$