Theorem permsetex

Description: The set of permutations of a set A exists. (Contributed by AV, 30-Mar-2024)

	Ref	Expression
Assertion	permsetex	$$⊢A\in V\to \left\{f|f:A\underset{1-1\; onto}{⟶}A\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		mapex	$$⊢\left(A\in V\wedge A\in V\right)\to \left\{f|f:A⟶A\right\}\in \mathrm{V}$$
2	1	anidms	$$⊢A\in V\to \left\{f|f:A⟶A\right\}\in \mathrm{V}$$
3		f1of	$$⊢f:A\underset{1-1\; onto}{⟶}A\to f:A⟶A$$
4	3	ss2abi	$$⊢\left\{f|f:A\underset{1-1\; onto}{⟶}A\right\}\subseteq \left\{f|f:A⟶A\right\}$$
5	4	a1i	$$⊢A\in V\to \left\{f|f:A\underset{1-1\; onto}{⟶}A\right\}\subseteq \left\{f|f:A⟶A\right\}$$
6	2 5	ssexd	$$⊢A\in V\to \left\{f|f:A\underset{1-1\; onto}{⟶}A\right\}\in \mathrm{V}$$