Theorem fndmdifeq0

Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015)

	Ref	Expression
Assertion	fndmdifeq0	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( dom ( 𝐹 ∖ 𝐺 ) = ∅ ↔ 𝐹 = 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		fndmdif	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → dom ( 𝐹 ∖ 𝐺 ) = { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) } )
2	1	eqeq1d	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( dom ( 𝐹 ∖ 𝐺 ) = ∅ ↔ { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) } = ∅ ) )
3		rabeq0	⊢ ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) } = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ¬ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) )
4		nne	⊢ ( ¬ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
5	4	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
6	3 5	bitri	⊢ ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) } = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
7		eqfnfv	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) )
8	6 7	bitr4id	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( { 𝑥 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐺 ‘ 𝑥 ) } = ∅ ↔ 𝐹 = 𝐺 ) )
9	2 8	bitrd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( dom ( 𝐹 ∖ 𝐺 ) = ∅ ↔ 𝐹 = 𝐺 ) )