Theorem dffun2OLD

Description: Obsolete version of dffun2 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996) Avoid ax-10 , ax-12 . (Revised by SN, 19-Dec-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dffun2OLD	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-fun	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ( 𝐴 ∘ ◡ 𝐴 ) ⊆ I ) )
2		cotrg	⊢ ( ( 𝐴 ∘ ◡ 𝐴 ) ⊆ I ↔ ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) )
3		alcom	⊢ ( ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) )
4		vex	⊢ 𝑦 ∈ V
5		vex	⊢ 𝑥 ∈ V
6	4 5	brcnv	⊢ ( 𝑦 ◡ 𝐴 𝑥 ↔ 𝑥 𝐴 𝑦 )
7	6	anbi1i	⊢ ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) ↔ ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) )
8		vex	⊢ 𝑧 ∈ V
9	8	ideq	⊢ ( 𝑦 I 𝑧 ↔ 𝑦 = 𝑧 )
10	7 9	imbi12i	⊢ ( ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) ↔ ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) )
11	10	3albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) )
12	3 11	bitri	⊢ ( ∀ 𝑦 ∀ 𝑥 ∀ 𝑧 ( ( 𝑦 ◡ 𝐴 𝑥 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 I 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) )
13	2 12	bitri	⊢ ( ( 𝐴 ∘ ◡ 𝐴 ) ⊆ I ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) )
14	13	anbi2i	⊢ ( ( Rel 𝐴 ∧ ( 𝐴 ∘ ◡ 𝐴 ) ⊆ I ) ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )
15	1 14	bitri	⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐴 𝑦 ∧ 𝑥 𝐴 𝑧 ) → 𝑦 = 𝑧 ) ) )