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Metamath Proof Explorer

Theorem ressbas

Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014) (Proof shortened by AV, 7-Nov-2024)

Ref Expression
Hypotheses ressbas.r ⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
ressbas.b ⊢ 𝐵 = ( Base ‘ 𝑊 )
Assertion ressbas ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 ressbas.r ⊢ 𝑅 = ( 𝑊 ↾s 𝐴 )
2 ressbas.b ⊢ 𝐵 = ( Base ‘ 𝑊 )
3 simp1 ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝐵 ⊆ 𝐴 )
4 sseqin2 ⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐵 )
5 3 4 sylib ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = 𝐵 )
6 1 2 ressid2 ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝑅 = 𝑊 )
7 6 fveq2d ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑊 ) )
8 2 5 7 3eqtr4a ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
9 8 3expib ⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) ) )
10 simp2 ⊢ ( ( ¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝑊 ∈ V )
11 2 fvexi ⊢ 𝐵 ∈ V
12 11 inex2 ⊢ ( 𝐴 ∩ 𝐵 ) ∈ V
13 baseid ⊢ Base = Slot ( Base ‘ ndx )
14 13 setsid ⊢ ( ( 𝑊 ∈ V ∧ ( 𝐴 ∩ 𝐵 ) ∈ V ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ 𝐵 ) ⟩ ) ) )
15 10 12 14 sylancl ⊢ ( ( ¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ 𝐵 ) ⟩ ) ) )
16 1 2 ressval2 ⊢ ( ( ¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝑅 = ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ 𝐵 ) ⟩ ) )
17 16 fveq2d ⊢ ( ( ¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( Base ‘ 𝑅 ) = ( Base ‘ ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ 𝐵 ) ⟩ ) ) )
18 15 17 eqtr4d ⊢ ( ( ¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
19 18 3expib ⊢ ( ¬ 𝐵 ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) ) )
20 9 19 pm2.61i ⊢ ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
21 in0 ⊢ ( 𝐴 ∩ ∅ ) = ∅
22 fvprc ⊢ ( ¬ 𝑊 ∈ V → ( Base ‘ 𝑊 ) = ∅ )
23 2 22 eqtrid ⊢ ( ¬ 𝑊 ∈ V → 𝐵 = ∅ )
24 23 ineq2d ⊢ ( ¬ 𝑊 ∈ V → ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∩ ∅ ) )
25 21 24 22 3eqtr4a ⊢ ( ¬ 𝑊 ∈ V → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑊 ) )
26 base0 ⊢ ∅ = ( Base ‘ ∅ )
27 26 eqcomi ⊢ ( Base ‘ ∅ ) = ∅
28 reldmress ⊢ Rel dom ↾s
29 27 1 28 oveqprc ⊢ ( ¬ 𝑊 ∈ V → ( Base ‘ 𝑊 ) = ( Base ‘ 𝑅 ) )
30 25 29 eqtrd ⊢ ( ¬ 𝑊 ∈ V → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
31 30 adantr ⊢ ( ( ¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )
32 20 31 pm2.61ian ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ 𝐵 ) = ( Base ‘ 𝑅 ) )