Step |
Hyp |
Ref |
Expression |
1 |
|
cnsscnp.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
3 |
1 2
|
cnf |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : 𝑋 ⟶ ∪ 𝐾 ) |
4 |
3
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐹 : 𝑋 ⟶ ∪ 𝐾 ) |
5 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝑦 ∈ 𝐾 ) → ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) |
6 |
5
|
ad2ant2r |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) |
7 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 ) |
8 |
7
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → 𝐴 ∈ 𝑋 ) |
9 |
|
simprr |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) |
10 |
3
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → 𝐹 : 𝑋 ⟶ ∪ 𝐾 ) |
11 |
|
ffn |
⊢ ( 𝐹 : 𝑋 ⟶ ∪ 𝐾 → 𝐹 Fn 𝑋 ) |
12 |
|
elpreima |
⊢ ( 𝐹 Fn 𝑋 → ( 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) ) |
13 |
10 11 12
|
3syl |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → ( 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) ) |
14 |
8 9 13
|
mpbir2and |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ) |
15 |
|
eqimss |
⊢ ( 𝑥 = ( ◡ 𝐹 “ 𝑦 ) → 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) |
16 |
15
|
biantrud |
⊢ ( 𝑥 = ( ◡ 𝐹 “ 𝑦 ) → ( 𝐴 ∈ 𝑥 ↔ ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) ) |
17 |
|
eleq2 |
⊢ ( 𝑥 = ( ◡ 𝐹 “ 𝑦 ) → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ) ) |
18 |
16 17
|
bitr3d |
⊢ ( 𝑥 = ( ◡ 𝐹 “ 𝑦 ) → ( ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ↔ 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ) ) |
19 |
18
|
rspcev |
⊢ ( ( ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ∧ 𝐴 ∈ ( ◡ 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) |
20 |
6 14 19
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 ) ) → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) |
21 |
20
|
expr |
⊢ ( ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) ) |
22 |
21
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) ) |
23 |
|
cntop1 |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ Top ) |
24 |
23
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐽 ∈ Top ) |
25 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
26 |
24 25
|
sylib |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
27 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐾 ∈ Top ) |
28 |
27
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐾 ∈ Top ) |
29 |
2
|
toptopon |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
30 |
28 29
|
sylib |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
31 |
|
iscnp3 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ↔ ( 𝐹 : 𝑋 ⟶ ∪ 𝐾 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) ) ) ) |
32 |
26 30 7 31
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ↔ ( 𝐹 : 𝑋 ⟶ ∪ 𝐾 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ ( ◡ 𝐹 “ 𝑦 ) ) ) ) ) ) |
33 |
4 22 32
|
mpbir2and |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) |