Theorem dffun3

Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996)

	Ref	Expression
Assertion	dffun3	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dffun2	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )
2		breq2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 𝐴 𝑦 ↔ 𝑥 𝐴 𝑧 ) )
3	2	mo4	⊢ ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) )
4		df-mo	⊢ ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) )
5	3 4	bitr3i	⊢ ( ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) )
6	5	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ↔ ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) )
7	6	anbi2i	⊢ ( ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) ) )
8	1 7	bitri	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝑥 𝐴 𝑦 → 𝑦 = 𝑧 ) ) )