Theorem fodmrnu

Description: An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006)

	Ref	Expression
Assertion	fodmrnu	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		fofn	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 Fn 𝐴 )
2		fofn	⊢ ( 𝐹 : 𝐶 –onto→ 𝐷 → 𝐹 Fn 𝐶 )
3		fndmu	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐶 ) → 𝐴 = 𝐶 )
4	1 2 3	syl2an	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐷 ) → 𝐴 = 𝐶 )
5		forn	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ran 𝐹 = 𝐵 )
6		forn	⊢ ( 𝐹 : 𝐶 –onto→ 𝐷 → ran 𝐹 = 𝐷 )
7	5 6	sylan9req	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐷 ) → 𝐵 = 𝐷 )
8	4 7	jca	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) )