Metamath Proof Explorer


Theorem resvlem

Description: Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018) (Revised by AV, 31-Oct-2024)

Ref Expression
Hypotheses resvlem.r 𝑅 = ( 𝑊v 𝐴 )
resvlem.e 𝐶 = ( 𝐸𝑊 )
resvlem.f 𝐸 = Slot ( 𝐸 ‘ ndx )
resvlem.n ( 𝐸 ‘ ndx ) ≠ ( Scalar ‘ ndx )
Assertion resvlem ( 𝐴𝑉𝐶 = ( 𝐸𝑅 ) )

Proof

Step Hyp Ref Expression
1 resvlem.r 𝑅 = ( 𝑊v 𝐴 )
2 resvlem.e 𝐶 = ( 𝐸𝑊 )
3 resvlem.f 𝐸 = Slot ( 𝐸 ‘ ndx )
4 resvlem.n ( 𝐸 ‘ ndx ) ≠ ( Scalar ‘ ndx )
5 eqid ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 eqid ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
7 1 5 6 resvid2 ( ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉 ) → 𝑅 = 𝑊 )
8 7 fveq2d ( ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
9 8 3expib ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) ) )
10 1 5 6 resvval2 ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉 ) → 𝑅 = ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) ⟩ ) )
11 10 fveq2d ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸 ‘ ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) ⟩ ) ) )
12 3 4 setsnid ( 𝐸𝑊 ) = ( 𝐸 ‘ ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) ⟩ ) )
13 11 12 eqtr4di ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
14 13 3expib ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) ) )
15 9 14 pm2.61i ( ( 𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
16 3 str0 ∅ = ( 𝐸 ‘ ∅ )
17 16 eqcomi ( 𝐸 ‘ ∅ ) = ∅
18 reldmresv Rel dom ↾v
19 17 1 18 oveqprc ( ¬ 𝑊 ∈ V → ( 𝐸𝑊 ) = ( 𝐸𝑅 ) )
20 19 eqcomd ( ¬ 𝑊 ∈ V → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
21 20 adantr ( ( ¬ 𝑊 ∈ V ∧ 𝐴𝑉 ) → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
22 15 21 pm2.61ian ( 𝐴𝑉 → ( 𝐸𝑅 ) = ( 𝐸𝑊 ) )
23 2 22 eqtr4id ( 𝐴𝑉𝐶 = ( 𝐸𝑅 ) )