Theorem elsymgbas

Description: Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008) (Revised by Mario Carneiro, 28-Jan-2015)

	Ref	Expression
Hypotheses	symgbas.1	$$⊢G=\mathrm{SymGrp}\left(A\right)$$
	symgbas.2	$$⊢B={\mathrm{Base}}_G$$
Assertion	elsymgbas	$$⊢A\in V\to \left(F\in B\leftrightarrow F:A\underset{1-1\; onto}{⟶}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	$$⊢G=\mathrm{SymGrp}\left(A\right)$$
2		symgbas.2	$$⊢B={\mathrm{Base}}_G$$
3		elex	$$⊢F\in B\to F\in \mathrm{V}$$
4	3	a1i	$$⊢A\in V\to \left(F\in B\to F\in \mathrm{V}\right)$$
5		f1of	$$⊢F:A\underset{1-1\; onto}{⟶}A\to F:A⟶A$$
6		fex	$$⊢\left(F:A⟶A\wedge A\in V\right)\to F\in \mathrm{V}$$
7	6	expcom	$$⊢A\in V\to \left(F:A⟶A\to F\in \mathrm{V}\right)$$
8	5 7	syl5	$$⊢A\in V\to \left(F:A\underset{1-1\; onto}{⟶}A\to F\in \mathrm{V}\right)$$
9	1 2	elsymgbas2	$$⊢F\in \mathrm{V}\to \left(F\in B\leftrightarrow F:A\underset{1-1\; onto}{⟶}A\right)$$
10	9	a1i	$$⊢A\in V\to \left(F\in \mathrm{V}\to \left(F\in B\leftrightarrow F:A\underset{1-1\; onto}{⟶}A\right)\right)$$
11	4 8 10	pm5.21ndd	$$⊢A\in V\to \left(F\in B\leftrightarrow F:A\underset{1-1\; onto}{⟶}A\right)$$