Theorem eqfnfvd

Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014)

	Ref	Expression
Hypotheses	eqfnfvd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
	eqfnfvd.2	⊢ ( 𝜑 → 𝐺 Fn 𝐴 )
	eqfnfvd.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
Assertion	eqfnfvd	⊢ ( 𝜑 → 𝐹 = 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		eqfnfvd.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2		eqfnfvd.2	⊢ ( 𝜑 → 𝐺 Fn 𝐴 )
3		eqfnfvd.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
4	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
5		eqfnfv	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) )
6	1 2 5	syl2anc	⊢ ( 𝜑 → ( 𝐹 = 𝐺 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) )
7	4 6	mpbird	⊢ ( 𝜑 → 𝐹 = 𝐺 )