Theorem fununiq

Description: The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013)

	Ref	Expression
Hypotheses	fununiq.1	⊢ 𝐴 ∈ V
	fununiq.2	⊢ 𝐵 ∈ V
	fununiq.3	⊢ 𝐶 ∈ V
Assertion	fununiq	⊢ ( Fun 𝐹 → ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fununiq.1	⊢ 𝐴 ∈ V
2		fununiq.2	⊢ 𝐵 ∈ V
3		fununiq.3	⊢ 𝐶 ∈ V
4		dffun2	⊢ ( Fun 𝐹 ↔ ( Rel 𝐹 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐹 𝑦 ∧ 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ) )
5		breq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 𝐹 𝑦 ↔ 𝐴 𝐹 𝐵 ) )
6	5	3adant3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑥 𝐹 𝑦 ↔ 𝐴 𝐹 𝐵 ) )
7		breq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑧 = 𝐶 ) → ( 𝑥 𝐹 𝑧 ↔ 𝐴 𝐹 𝐶 ) )
8	7	3adant2	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑥 𝐹 𝑧 ↔ 𝐴 𝐹 𝐶 ) )
9	6 8	anbi12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( 𝑥 𝐹 𝑦 ∧ 𝑥 𝐹 𝑧 ) ↔ ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) ) )
10		eqeq12	⊢ ( ( 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑦 = 𝑧 ↔ 𝐵 = 𝐶 ) )
11	10	3adant1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑦 = 𝑧 ↔ 𝐵 = 𝐶 ) )
12	9 11	imbi12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( ( 𝑥 𝐹 𝑦 ∧ 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) )
13	12	spc3gv	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V ) → ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐹 𝑦 ∧ 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) → ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) )
14	1 2 3 13	mp3an	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐹 𝑦 ∧ 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) → ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )
15	4 14	simplbiim	⊢ ( Fun 𝐹 → ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )