Theorem funbreq

Description: An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013)

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

Proof

Step	Hyp	Ref	Expression
1		funbreq.1	⊢ 𝐴 ∈ V
2		funbreq.2	⊢ 𝐵 ∈ V
3		funbreq.3	⊢ 𝐶 ∈ V
4	1 2 3	fununiq	⊢ ( Fun 𝐹 → ( ( 𝐴 𝐹 𝐵 ∧ 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )
5	4	expdimp	⊢ ( ( Fun 𝐹 ∧ 𝐴 𝐹 𝐵 ) → ( 𝐴 𝐹 𝐶 → 𝐵 = 𝐶 ) )
6		breq2	⊢ ( 𝐵 = 𝐶 → ( 𝐴 𝐹 𝐵 ↔ 𝐴 𝐹 𝐶 ) )
7	6	biimpcd	⊢ ( 𝐴 𝐹 𝐵 → ( 𝐵 = 𝐶 → 𝐴 𝐹 𝐶 ) )
8	7	adantl	⊢ ( ( Fun 𝐹 ∧ 𝐴 𝐹 𝐵 ) → ( 𝐵 = 𝐶 → 𝐴 𝐹 𝐶 ) )
9	5 8	impbid	⊢ ( ( Fun 𝐹 ∧ 𝐴 𝐹 𝐵 ) → ( 𝐴 𝐹 𝐶 ↔ 𝐵 = 𝐶 ) )