Metamath Proof Explorer


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 𝐴𝐺 Fn 𝐴 ) → ( dom ( 𝐹𝐺 ) = ∅ ↔ 𝐹 = 𝐺 ) )

Proof

Step Hyp Ref Expression
1 fndmdif ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → dom ( 𝐹𝐺 ) = { 𝑥𝐴 ∣ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) } )
2 1 eqeq1d ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( dom ( 𝐹𝐺 ) = ∅ ↔ { 𝑥𝐴 ∣ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) } = ∅ ) )
3 rabeq0 ( { 𝑥𝐴 ∣ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) } = ∅ ↔ ∀ 𝑥𝐴 ¬ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) )
4 nne ( ¬ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) ↔ ( 𝐹𝑥 ) = ( 𝐺𝑥 ) )
5 4 ralbii ( ∀ 𝑥𝐴 ¬ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) )
6 3 5 bitri ( { 𝑥𝐴 ∣ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) } = ∅ ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) )
7 eqfnfv ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ∀ 𝑥𝐴 ( 𝐹𝑥 ) = ( 𝐺𝑥 ) ) )
8 6 7 bitr4id ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( { 𝑥𝐴 ∣ ( 𝐹𝑥 ) ≠ ( 𝐺𝑥 ) } = ∅ ↔ 𝐹 = 𝐺 ) )
9 2 8 bitrd ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( dom ( 𝐹𝐺 ) = ∅ ↔ 𝐹 = 𝐺 ) )