Metamath Proof Explorer


Theorem fnresin

Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017)

Ref Expression
Assertion fnresin ( 𝐹 Fn 𝐴 → ( 𝐹𝐵 ) Fn ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 fnresin1 ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐴𝐵 ) ) Fn ( 𝐴𝐵 ) )
2 resindi ( 𝐹 ↾ ( 𝐴𝐵 ) ) = ( ( 𝐹𝐴 ) ∩ ( 𝐹𝐵 ) )
3 fnresdm ( 𝐹 Fn 𝐴 → ( 𝐹𝐴 ) = 𝐹 )
4 3 ineq1d ( 𝐹 Fn 𝐴 → ( ( 𝐹𝐴 ) ∩ ( 𝐹𝐵 ) ) = ( 𝐹 ∩ ( 𝐹𝐵 ) ) )
5 incom ( ( 𝐹𝐵 ) ∩ 𝐹 ) = ( 𝐹 ∩ ( 𝐹𝐵 ) )
6 resss ( 𝐹𝐵 ) ⊆ 𝐹
7 df-ss ( ( 𝐹𝐵 ) ⊆ 𝐹 ↔ ( ( 𝐹𝐵 ) ∩ 𝐹 ) = ( 𝐹𝐵 ) )
8 6 7 mpbi ( ( 𝐹𝐵 ) ∩ 𝐹 ) = ( 𝐹𝐵 )
9 5 8 eqtr3i ( 𝐹 ∩ ( 𝐹𝐵 ) ) = ( 𝐹𝐵 )
10 4 9 eqtrdi ( 𝐹 Fn 𝐴 → ( ( 𝐹𝐴 ) ∩ ( 𝐹𝐵 ) ) = ( 𝐹𝐵 ) )
11 2 10 syl5eq ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐴𝐵 ) ) = ( 𝐹𝐵 ) )
12 11 fneq1d ( 𝐹 Fn 𝐴 → ( ( 𝐹 ↾ ( 𝐴𝐵 ) ) Fn ( 𝐴𝐵 ) ↔ ( 𝐹𝐵 ) Fn ( 𝐴𝐵 ) ) )
13 1 12 mpbid ( 𝐹 Fn 𝐴 → ( 𝐹𝐵 ) Fn ( 𝐴𝐵 ) )