Theorem symgextf1o

Description: The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019)

	Ref	Expression
Hypotheses	symgext.s	$$⊢S={\mathrm{Base}}_{\mathrm{SymGrp}\left(N\setminus \left\{K\right\}\right)}$$
	symgext.e	$$⊢E=\left(x\in N⟼if\left(x=K,K,Z\left(x\right)\right)\right)$$
Assertion	symgextf1o	$$⊢\left(K\in N\wedge Z\in S\right)\to E:N\underset{1-1\; onto}{⟶}N$$

Proof

Step	Hyp	Ref	Expression
1		symgext.s	$$⊢S={\mathrm{Base}}_{\mathrm{SymGrp}\left(N\setminus \left\{K\right\}\right)}$$
2		symgext.e	$$⊢E=\left(x\in N⟼if\left(x=K,K,Z\left(x\right)\right)\right)$$
3	1 2	symgextf1	$$⊢\left(K\in N\wedge Z\in S\right)\to E:N\underset{1-1}{⟶}N$$
4	1 2	symgextfo	$$⊢\left(K\in N\wedge Z\in S\right)\to E:N\underset{onto}{⟶}N$$
5		df-f1o	$$⊢E:N\underset{1-1\; onto}{⟶}N\leftrightarrow \left(E:N\underset{1-1}{⟶}N\wedge E:N\underset{onto}{⟶}N\right)$$
6	3 4 5	sylanbrc	$$⊢\left(K\in N\wedge Z\in S\right)\to E:N\underset{1-1\; onto}{⟶}N$$