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