Metamath Proof Explorer


Theorem fnunres2

Description: Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023)

Ref Expression
Assertion fnunres2 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( ( 𝐹𝐺 ) ↾ 𝐵 ) = 𝐺 )

Proof

Step Hyp Ref Expression
1 uncom ( 𝐹𝐺 ) = ( 𝐺𝐹 )
2 1 reseq1i ( ( 𝐹𝐺 ) ↾ 𝐵 ) = ( ( 𝐺𝐹 ) ↾ 𝐵 )
3 simp2 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → 𝐺 Fn 𝐵 )
4 simp1 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → 𝐹 Fn 𝐴 )
5 incom ( 𝐴𝐵 ) = ( 𝐵𝐴 )
6 simp3 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( 𝐴𝐵 ) = ∅ )
7 5 6 eqtr3id ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( 𝐵𝐴 ) = ∅ )
8 fnunres1 ( ( 𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ ( 𝐵𝐴 ) = ∅ ) → ( ( 𝐺𝐹 ) ↾ 𝐵 ) = 𝐺 )
9 3 4 7 8 syl3anc ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( ( 𝐺𝐹 ) ↾ 𝐵 ) = 𝐺 )
10 2 9 syl5eq ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ( 𝐴𝐵 ) = ∅ ) → ( ( 𝐹𝐺 ) ↾ 𝐵 ) = 𝐺 )