Step |
Hyp |
Ref |
Expression |
1 |
|
isnacs.f |
⊢ 𝐹 = ( mrCls ‘ 𝐶 ) |
2 |
|
elfvex |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → 𝑋 ∈ V ) |
3 |
|
elfvex |
⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → 𝑋 ∈ V ) |
4 |
3
|
adantr |
⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) → 𝑋 ∈ V ) |
5 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( ACS ‘ 𝑥 ) = ( ACS ‘ 𝑋 ) ) |
6 |
|
pweq |
⊢ ( 𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋 ) |
7 |
6
|
ineq1d |
⊢ ( 𝑥 = 𝑋 → ( 𝒫 𝑥 ∩ Fin ) = ( 𝒫 𝑋 ∩ Fin ) ) |
8 |
7
|
rexeqdv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑔 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ) ) |
9 |
8
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ↔ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ) ) |
10 |
5 9
|
rabeqbidv |
⊢ ( 𝑥 = 𝑋 → { 𝑐 ∈ ( ACS ‘ 𝑥 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } = { 𝑐 ∈ ( ACS ‘ 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ) |
11 |
|
df-nacs |
⊢ NoeACS = ( 𝑥 ∈ V ↦ { 𝑐 ∈ ( ACS ‘ 𝑥 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ) |
12 |
|
fvex |
⊢ ( ACS ‘ 𝑋 ) ∈ V |
13 |
12
|
rabex |
⊢ { 𝑐 ∈ ( ACS ‘ 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ∈ V |
14 |
10 11 13
|
fvmpt |
⊢ ( 𝑋 ∈ V → ( NoeACS ‘ 𝑋 ) = { 𝑐 ∈ ( ACS ‘ 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ) |
15 |
14
|
eleq2d |
⊢ ( 𝑋 ∈ V → ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ↔ 𝐶 ∈ { 𝑐 ∈ ( ACS ‘ 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ) ) |
16 |
|
fveq2 |
⊢ ( 𝑐 = 𝐶 → ( mrCls ‘ 𝑐 ) = ( mrCls ‘ 𝐶 ) ) |
17 |
16 1
|
eqtr4di |
⊢ ( 𝑐 = 𝐶 → ( mrCls ‘ 𝑐 ) = 𝐹 ) |
18 |
17
|
fveq1d |
⊢ ( 𝑐 = 𝐶 → ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) = ( 𝐹 ‘ 𝑔 ) ) |
19 |
18
|
eqeq2d |
⊢ ( 𝑐 = 𝐶 → ( 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ↔ 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) |
20 |
19
|
rexbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) |
21 |
20
|
raleqbi1dv |
⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) ↔ ∀ 𝑠 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) |
22 |
21
|
elrab |
⊢ ( 𝐶 ∈ { 𝑐 ∈ ( ACS ‘ 𝑋 ) ∣ ∀ 𝑠 ∈ 𝑐 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( ( mrCls ‘ 𝑐 ) ‘ 𝑔 ) } ↔ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) |
23 |
15 22
|
bitrdi |
⊢ ( 𝑋 ∈ V → ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) ) |
24 |
2 4 23
|
pm5.21nii |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑠 = ( 𝐹 ‘ 𝑔 ) ) ) |