Metamath Proof Explorer


Theorem neicvgbex

Description: If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, H , then the base set exists. (Contributed by RP, 4-Jun-2021)

Ref Expression
Hypotheses neicvgbex.d 𝐷 = ( 𝑃𝐵 )
neicvgbex.h 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) )
neicvgbex.r ( 𝜑𝑁 𝐻 𝑀 )
Assertion neicvgbex ( 𝜑𝐵 ∈ V )

Proof

Step Hyp Ref Expression
1 neicvgbex.d 𝐷 = ( 𝑃𝐵 )
2 neicvgbex.h 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) )
3 neicvgbex.r ( 𝜑𝑁 𝐻 𝑀 )
4 1 coeq1i ( 𝐷𝐺 ) = ( ( 𝑃𝐵 ) ∘ 𝐺 )
5 4 coeq2i ( 𝐹 ∘ ( 𝐷𝐺 ) ) = ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) )
6 2 5 eqtri 𝐻 = ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) )
7 6 a1i ( 𝜑𝐻 = ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) )
8 7 3 breqdi ( 𝜑𝑁 ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) 𝑀 )
9 brne0 ( 𝑁 ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) 𝑀 → ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) ≠ ∅ )
10 fvprc ( ¬ 𝐵 ∈ V → ( 𝑃𝐵 ) = ∅ )
11 10 dmeqd ( ¬ 𝐵 ∈ V → dom ( 𝑃𝐵 ) = dom ∅ )
12 dm0 dom ∅ = ∅
13 11 12 eqtrdi ( ¬ 𝐵 ∈ V → dom ( 𝑃𝐵 ) = ∅ )
14 13 ineq1d ( ¬ 𝐵 ∈ V → ( dom ( 𝑃𝐵 ) ∩ ran 𝐺 ) = ( ∅ ∩ ran 𝐺 ) )
15 0in ( ∅ ∩ ran 𝐺 ) = ∅
16 14 15 eqtrdi ( ¬ 𝐵 ∈ V → ( dom ( 𝑃𝐵 ) ∩ ran 𝐺 ) = ∅ )
17 16 coemptyd ( ¬ 𝐵 ∈ V → ( ( 𝑃𝐵 ) ∘ 𝐺 ) = ∅ )
18 17 rneqd ( ¬ 𝐵 ∈ V → ran ( ( 𝑃𝐵 ) ∘ 𝐺 ) = ran ∅ )
19 rn0 ran ∅ = ∅
20 18 19 eqtrdi ( ¬ 𝐵 ∈ V → ran ( ( 𝑃𝐵 ) ∘ 𝐺 ) = ∅ )
21 20 ineq2d ( ¬ 𝐵 ∈ V → ( dom 𝐹 ∩ ran ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) = ( dom 𝐹 ∩ ∅ ) )
22 in0 ( dom 𝐹 ∩ ∅ ) = ∅
23 21 22 eqtrdi ( ¬ 𝐵 ∈ V → ( dom 𝐹 ∩ ran ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) = ∅ )
24 23 coemptyd ( ¬ 𝐵 ∈ V → ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) = ∅ )
25 24 necon1ai ( ( 𝐹 ∘ ( ( 𝑃𝐵 ) ∘ 𝐺 ) ) ≠ ∅ → 𝐵 ∈ V )
26 8 9 25 3syl ( 𝜑𝐵 ∈ V )