Theorem dffun8

Description: Alternate definition of a function. One possibility for the definition of a function in Enderton p. 42. Compare dffun7 . (Contributed by NM, 4-Nov-2002) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	dffun8	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		dffun7	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ) )
2		moeu	⊢ ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) )
3		vex	⊢ 𝑥 ∈ V
4	3	eldm	⊢ ( 𝑥 ∈ dom 𝐴 ↔ ∃ 𝑦 𝑥 𝐴 𝑦 )
5		pm5.5	⊢ ( ∃ 𝑦 𝑥 𝐴 𝑦 → ( ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) )
6	4 5	sylbi	⊢ ( 𝑥 ∈ dom 𝐴 → ( ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) )
7	2 6	syl5bb	⊢ ( 𝑥 ∈ dom 𝐴 → ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) )
8	7	ralbiia	⊢ ( ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 )
9	8	anbi2i	⊢ ( ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ) ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) )
10	1 9	bitri	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) )