Theorem hasheqf1o

Description: The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017)

	Ref	Expression
Assertion	hasheqf1o	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		hashen	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
2		bren	⊢ ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 )
3	1 2	bitrdi	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )