Step |
Hyp |
Ref |
Expression |
1 |
|
smfpimcclem.n |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
smfpimcclem.z |
⊢ 𝑍 ∈ 𝑉 |
3 |
|
smfpimcclem.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑊 ) |
4 |
|
smfpimcclem.c |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) → ( 𝐶 ‘ 𝑦 ) ∈ 𝑦 ) |
5 |
|
smfpimcclem.h |
⊢ 𝐻 = ( 𝑛 ∈ 𝑍 ↦ ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
6 |
|
nfcv |
⊢ Ⅎ 𝑠 𝑆 |
7 |
6
|
ssrab2f |
⊢ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ⊆ 𝑆 |
8 |
|
eqid |
⊢ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } |
9 |
8 3
|
rabexd |
⊢ ( 𝜑 → { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ V ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ V ) |
11 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝜑 ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
13 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) = ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) |
14 |
13
|
elrnmpt1 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ V ) → { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
15 |
12 10 14
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
16 |
11 15
|
jca |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝜑 ∧ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ) |
17 |
|
eleq1 |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → ( 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ↔ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ↔ ( 𝜑 ∧ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → ( 𝐶 ‘ 𝑦 ) = ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
20 |
|
id |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) |
21 |
19 20
|
eleq12d |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → ( ( 𝐶 ‘ 𝑦 ) ∈ 𝑦 ↔ ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
22 |
18 21
|
imbi12d |
⊢ ( 𝑦 = { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } → ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) → ( 𝐶 ‘ 𝑦 ) ∈ 𝑦 ) ↔ ( ( 𝜑 ∧ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) → ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ) |
23 |
22 4
|
vtoclg |
⊢ ( { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ V → ( ( 𝜑 ∧ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ∈ ran ( 𝑛 ∈ 𝑍 ↦ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) → ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
24 |
10 16 23
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) |
25 |
7 24
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ 𝑆 ) |
26 |
1 25 5
|
fmptdf |
⊢ ( 𝜑 → 𝐻 : 𝑍 ⟶ 𝑆 ) |
27 |
|
nfcv |
⊢ Ⅎ 𝑠 𝐶 |
28 |
|
nfrab1 |
⊢ Ⅎ 𝑠 { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } |
29 |
27 28
|
nffv |
⊢ Ⅎ 𝑠 ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) |
30 |
|
nfcv |
⊢ Ⅎ 𝑠 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) |
31 |
|
nfcv |
⊢ Ⅎ 𝑠 dom ( 𝐹 ‘ 𝑛 ) |
32 |
29 31
|
nfin |
⊢ Ⅎ 𝑠 ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) |
33 |
30 32
|
nfeq |
⊢ Ⅎ 𝑠 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) |
34 |
|
ineq1 |
⊢ ( 𝑠 = ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) → ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
35 |
34
|
eqeq2d |
⊢ ( 𝑠 = ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) → ( ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) ↔ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
36 |
29 6 33 35
|
elrabf |
⊢ ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ↔ ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ 𝑆 ∧ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
37 |
24 36
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ 𝑆 ∧ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
38 |
37
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
39 |
5
|
a1i |
⊢ ( 𝜑 → 𝐻 = ( 𝑛 ∈ 𝑍 ↦ ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ) |
40 |
24
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∈ V ) |
41 |
39 40
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐻 ‘ 𝑛 ) = ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
42 |
41
|
ineq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) = ( ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
43 |
38 42
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
44 |
43
|
ex |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 → ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
45 |
1 44
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
46 |
2
|
elexi |
⊢ 𝑍 ∈ V |
47 |
46
|
mptex |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) ∈ V |
48 |
5 47
|
eqeltri |
⊢ 𝐻 ∈ V |
49 |
|
feq1 |
⊢ ( ℎ = 𝐻 → ( ℎ : 𝑍 ⟶ 𝑆 ↔ 𝐻 : 𝑍 ⟶ 𝑆 ) ) |
50 |
|
nfcv |
⊢ Ⅎ 𝑛 ℎ |
51 |
|
nfmpt1 |
⊢ Ⅎ 𝑛 ( 𝑛 ∈ 𝑍 ↦ ( 𝐶 ‘ { 𝑠 ∈ 𝑆 ∣ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( 𝑠 ∩ dom ( 𝐹 ‘ 𝑛 ) ) } ) ) |
52 |
5 51
|
nfcxfr |
⊢ Ⅎ 𝑛 𝐻 |
53 |
50 52
|
nfeq |
⊢ Ⅎ 𝑛 ℎ = 𝐻 |
54 |
|
fveq1 |
⊢ ( ℎ = 𝐻 → ( ℎ ‘ 𝑛 ) = ( 𝐻 ‘ 𝑛 ) ) |
55 |
54
|
ineq1d |
⊢ ( ℎ = 𝐻 → ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) |
56 |
55
|
eqeq2d |
⊢ ( ℎ = 𝐻 → ( ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ↔ ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
57 |
53 56
|
ralbid |
⊢ ( ℎ = 𝐻 → ( ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ↔ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
58 |
49 57
|
anbi12d |
⊢ ( ℎ = 𝐻 → ( ( ℎ : 𝑍 ⟶ 𝑆 ∧ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ↔ ( 𝐻 : 𝑍 ⟶ 𝑆 ∧ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
59 |
48 58
|
spcev |
⊢ ( ( 𝐻 : 𝑍 ⟶ 𝑆 ∧ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( 𝐻 ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) → ∃ ℎ ( ℎ : 𝑍 ⟶ 𝑆 ∧ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |
60 |
26 45 59
|
syl2anc |
⊢ ( 𝜑 → ∃ ℎ ( ℎ : 𝑍 ⟶ 𝑆 ∧ ∀ 𝑛 ∈ 𝑍 ( ◡ ( 𝐹 ‘ 𝑛 ) “ 𝐴 ) = ( ( ℎ ‘ 𝑛 ) ∩ dom ( 𝐹 ‘ 𝑛 ) ) ) ) |