Theorem symghash

Description: The symmetric group on n objects has cardinality n ! . (Contributed by Mario Carneiro, 22-Jan-2015)

	Ref	Expression
Hypotheses	symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	symghash	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ! ‘ ( ♯ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1 2	symgbas	⊢ 𝐵 = { 𝑓 ∣ 𝑓 : 𝐴 –1-1-onto→ 𝐴 }
4	3	fveq2i	⊢ ( ♯ ‘ 𝐵 ) = ( ♯ ‘ { 𝑓 ∣ 𝑓 : 𝐴 –1-1-onto→ 𝐴 } )
5		hashfac	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ { 𝑓 ∣ 𝑓 : 𝐴 –1-1-onto→ 𝐴 } ) = ( ! ‘ ( ♯ ‘ 𝐴 ) ) )
6	4 5	eqtrid	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ! ‘ ( ♯ ‘ 𝐴 ) ) )