Step |
Hyp |
Ref |
Expression |
1 |
|
txflf.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
txflf.k |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
3 |
|
txflf.l |
⊢ ( 𝜑 → 𝐿 ∈ ( Fil ‘ 𝑍 ) ) |
4 |
|
txflf.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑋 ) |
5 |
|
txflf.g |
⊢ ( 𝜑 → 𝐺 : 𝑍 ⟶ 𝑌 ) |
6 |
|
txflf.h |
⊢ 𝐻 = ( 𝑛 ∈ 𝑍 ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) |
7 |
|
vex |
⊢ 𝑢 ∈ V |
8 |
|
vex |
⊢ 𝑣 ∈ V |
9 |
7 8
|
xpex |
⊢ ( 𝑢 × 𝑣 ) ∈ V |
10 |
9
|
rgen2w |
⊢ ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 𝑢 × 𝑣 ) ∈ V |
11 |
|
eqid |
⊢ ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) = ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) |
12 |
|
eleq2 |
⊢ ( 𝑧 = ( 𝑢 × 𝑣 ) → ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 ↔ 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) ) ) |
13 |
|
sseq2 |
⊢ ( 𝑧 = ( 𝑢 × 𝑣 ) → ( ( 𝐻 “ ℎ ) ⊆ 𝑧 ↔ ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ) |
14 |
13
|
rexbidv |
⊢ ( 𝑧 = ( 𝑢 × 𝑣 ) → ( ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ↔ ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ) |
15 |
12 14
|
imbi12d |
⊢ ( 𝑧 = ( 𝑢 × 𝑣 ) → ( ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ↔ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ) ) |
16 |
11 15
|
ralrnmpo |
⊢ ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 𝑢 × 𝑣 ) ∈ V → ( ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ↔ ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ) ) |
17 |
10 16
|
ax-mp |
⊢ ( ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ↔ ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ) |
18 |
|
opelxp |
⊢ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ( 𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣 ) ) |
19 |
18
|
biancomi |
⊢ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ( 𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢 ) ) |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ( 𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢 ) ) ) |
21 |
|
r19.40 |
⊢ ( ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ( ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
22 |
|
raleq |
⊢ ( ℎ = 𝑓 → ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ↔ ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
23 |
22
|
cbvrexvw |
⊢ ( ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ↔ ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) |
24 |
|
raleq |
⊢ ( ℎ = 𝑔 → ( ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ↔ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
25 |
24
|
cbvrexvw |
⊢ ( ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ↔ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) |
26 |
23 25
|
anbi12i |
⊢ ( ( ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ ℎ ∈ 𝐿 ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
27 |
21 26
|
sylib |
⊢ ( ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
28 |
|
reeanv |
⊢ ( ∃ 𝑓 ∈ 𝐿 ∃ 𝑔 ∈ 𝐿 ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
29 |
|
filin |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿 ) → ( 𝑓 ∩ 𝑔 ) ∈ 𝐿 ) |
30 |
29
|
3expb |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ ( 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿 ) ) → ( 𝑓 ∩ 𝑔 ) ∈ 𝐿 ) |
31 |
3 30
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿 ) ) → ( 𝑓 ∩ 𝑔 ) ∈ 𝐿 ) |
32 |
|
inss1 |
⊢ ( 𝑓 ∩ 𝑔 ) ⊆ 𝑓 |
33 |
|
ssralv |
⊢ ( ( 𝑓 ∩ 𝑔 ) ⊆ 𝑓 → ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 → ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
34 |
32 33
|
ax-mp |
⊢ ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 → ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) |
35 |
|
inss2 |
⊢ ( 𝑓 ∩ 𝑔 ) ⊆ 𝑔 |
36 |
|
ssralv |
⊢ ( ( 𝑓 ∩ 𝑔 ) ⊆ 𝑔 → ( ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 → ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
37 |
35 36
|
ax-mp |
⊢ ( ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 → ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) |
38 |
34 37
|
anim12i |
⊢ ( ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ( ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
39 |
|
raleq |
⊢ ( ℎ = ( 𝑓 ∩ 𝑔 ) → ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ↔ ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
40 |
|
raleq |
⊢ ( ℎ = ( 𝑓 ∩ 𝑔 ) → ( ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ↔ ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
41 |
39 40
|
anbi12d |
⊢ ( ℎ = ( 𝑓 ∩ 𝑔 ) → ( ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ↔ ( ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
42 |
41
|
rspcev |
⊢ ( ( ( 𝑓 ∩ 𝑔 ) ∈ 𝐿 ∧ ( ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ( 𝑓 ∩ 𝑔 ) ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) → ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
43 |
31 38 42
|
syl2an |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿 ) ) ∧ ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) → ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
44 |
43
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿 ) ) → ( ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
45 |
44
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝐿 ∃ 𝑔 ∈ 𝐿 ( ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
46 |
28 45
|
syl5bir |
⊢ ( 𝜑 → ( ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) → ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
47 |
27 46
|
impbid2 |
⊢ ( 𝜑 → ( ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
48 |
|
df-ima |
⊢ ( 𝐻 “ ℎ ) = ran ( 𝐻 ↾ ℎ ) |
49 |
|
filelss |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ ℎ ∈ 𝐿 ) → ℎ ⊆ 𝑍 ) |
50 |
3 49
|
sylan |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ℎ ⊆ 𝑍 ) |
51 |
6
|
reseq1i |
⊢ ( 𝐻 ↾ ℎ ) = ( ( 𝑛 ∈ 𝑍 ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ↾ ℎ ) |
52 |
|
resmpt |
⊢ ( ℎ ⊆ 𝑍 → ( ( 𝑛 ∈ 𝑍 ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ↾ ℎ ) = ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ) |
53 |
51 52
|
syl5eq |
⊢ ( ℎ ⊆ 𝑍 → ( 𝐻 ↾ ℎ ) = ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ) |
54 |
50 53
|
syl |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ( 𝐻 ↾ ℎ ) = ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ) |
55 |
54
|
rneqd |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ran ( 𝐻 ↾ ℎ ) = ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ) |
56 |
48 55
|
syl5eq |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ( 𝐻 “ ℎ ) = ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ) |
57 |
56
|
sseq1d |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ( ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ↔ ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ⊆ ( 𝑢 × 𝑣 ) ) ) |
58 |
|
opelxp |
⊢ ( 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ( ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
59 |
58
|
ralbii |
⊢ ( ∀ 𝑛 ∈ ℎ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ∀ 𝑛 ∈ ℎ ( ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
60 |
|
eqid |
⊢ ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) = ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) |
61 |
60
|
fmpt |
⊢ ( ∀ 𝑛 ∈ ℎ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) : ℎ ⟶ ( 𝑢 × 𝑣 ) ) |
62 |
|
opex |
⊢ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ V |
63 |
62 60
|
fnmpti |
⊢ ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) Fn ℎ |
64 |
|
df-f |
⊢ ( ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) : ℎ ⟶ ( 𝑢 × 𝑣 ) ↔ ( ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) Fn ℎ ∧ ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ⊆ ( 𝑢 × 𝑣 ) ) ) |
65 |
63 64
|
mpbiran |
⊢ ( ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) : ℎ ⟶ ( 𝑢 × 𝑣 ) ↔ ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ⊆ ( 𝑢 × 𝑣 ) ) |
66 |
61 65
|
bitri |
⊢ ( ∀ 𝑛 ∈ ℎ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ ( 𝑢 × 𝑣 ) ↔ ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ⊆ ( 𝑢 × 𝑣 ) ) |
67 |
|
r19.26 |
⊢ ( ∀ 𝑛 ∈ ℎ ( ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ↔ ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
68 |
59 66 67
|
3bitr3i |
⊢ ( ran ( 𝑛 ∈ ℎ ↦ 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ) ⊆ ( 𝑢 × 𝑣 ) ↔ ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
69 |
57 68
|
bitrdi |
⊢ ( ( 𝜑 ∧ ℎ ∈ 𝐿 ) → ( ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ↔ ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
70 |
69
|
rexbidva |
⊢ ( 𝜑 → ( ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ↔ ∃ ℎ ∈ 𝐿 ( ∀ 𝑛 ∈ ℎ ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∀ 𝑛 ∈ ℎ ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
71 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → 𝐹 : 𝑍 ⟶ 𝑋 ) |
72 |
71
|
ffund |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → Fun 𝐹 ) |
73 |
|
filelss |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝑓 ∈ 𝐿 ) → 𝑓 ⊆ 𝑍 ) |
74 |
3 73
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → 𝑓 ⊆ 𝑍 ) |
75 |
71
|
fdmd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → dom 𝐹 = 𝑍 ) |
76 |
74 75
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → 𝑓 ⊆ dom 𝐹 ) |
77 |
|
funimass4 |
⊢ ( ( Fun 𝐹 ∧ 𝑓 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ↔ ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
78 |
72 76 77
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐿 ) → ( ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ↔ ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
79 |
78
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ↔ ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ) ) |
80 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → 𝐺 : 𝑍 ⟶ 𝑌 ) |
81 |
80
|
ffund |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → Fun 𝐺 ) |
82 |
|
filelss |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝑔 ∈ 𝐿 ) → 𝑔 ⊆ 𝑍 ) |
83 |
3 82
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → 𝑔 ⊆ 𝑍 ) |
84 |
80
|
fdmd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → dom 𝐺 = 𝑍 ) |
85 |
83 84
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → 𝑔 ⊆ dom 𝐺 ) |
86 |
|
funimass4 |
⊢ ( ( Fun 𝐺 ∧ 𝑔 ⊆ dom 𝐺 ) → ( ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ↔ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
87 |
81 85 86
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐿 ) → ( ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ↔ ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
88 |
87
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ↔ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) |
89 |
79 88
|
anbi12d |
⊢ ( 𝜑 → ( ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ∀ 𝑛 ∈ 𝑓 ( 𝐹 ‘ 𝑛 ) ∈ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ∀ 𝑛 ∈ 𝑔 ( 𝐺 ‘ 𝑛 ) ∈ 𝑣 ) ) ) |
90 |
47 70 89
|
3bitr4d |
⊢ ( 𝜑 → ( ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
91 |
20 90
|
imbi12d |
⊢ ( 𝜑 → ( ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ( ( 𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢 ) → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
92 |
|
impexp |
⊢ ( ( ( 𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢 ) → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ↔ ( 𝑆 ∈ 𝑣 → ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
93 |
91 92
|
bitrdi |
⊢ ( 𝜑 → ( ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ( 𝑆 ∈ 𝑣 → ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) ) |
94 |
93
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) ) |
95 |
|
eleq2 |
⊢ ( 𝑥 = 𝑣 → ( 𝑆 ∈ 𝑥 ↔ 𝑆 ∈ 𝑣 ) ) |
96 |
95
|
ralrab |
⊢ ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ↔ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
97 |
|
r19.21v |
⊢ ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ↔ ( 𝑅 ∈ 𝑢 → ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
98 |
96 97
|
bitr3i |
⊢ ( ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ( 𝑅 ∈ 𝑢 → ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ↔ ( 𝑅 ∈ 𝑢 → ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
99 |
94 98
|
bitrdi |
⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ( 𝑅 ∈ 𝑢 → ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
100 |
99
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
101 |
|
eleq2 |
⊢ ( 𝑥 = 𝑢 → ( 𝑅 ∈ 𝑥 ↔ 𝑅 ∈ 𝑢 ) ) |
102 |
101
|
ralrab |
⊢ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
103 |
100 102
|
bitr4di |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
104 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
105 |
|
toponmax |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 ∈ 𝐽 ) |
106 |
1 105
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
107 |
|
eleq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑅 ∈ 𝑥 ↔ 𝑅 ∈ 𝑋 ) ) |
108 |
107
|
rspcev |
⊢ ( ( 𝑋 ∈ 𝐽 ∧ 𝑅 ∈ 𝑋 ) → ∃ 𝑥 ∈ 𝐽 𝑅 ∈ 𝑥 ) |
109 |
|
rabn0 |
⊢ ( { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐽 𝑅 ∈ 𝑥 ) |
110 |
108 109
|
sylibr |
⊢ ( ( 𝑋 ∈ 𝐽 ∧ 𝑅 ∈ 𝑋 ) → { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ ) |
111 |
106 110
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑅 ∈ 𝑋 ) → { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ ) |
112 |
|
toponmax |
⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) → 𝑌 ∈ 𝐾 ) |
113 |
2 112
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ 𝐾 ) |
114 |
|
eleq2 |
⊢ ( 𝑥 = 𝑌 → ( 𝑆 ∈ 𝑥 ↔ 𝑆 ∈ 𝑌 ) ) |
115 |
114
|
rspcev |
⊢ ( ( 𝑌 ∈ 𝐾 ∧ 𝑆 ∈ 𝑌 ) → ∃ 𝑥 ∈ 𝐾 𝑆 ∈ 𝑥 ) |
116 |
|
rabn0 |
⊢ ( { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐾 𝑆 ∈ 𝑥 ) |
117 |
115 116
|
sylibr |
⊢ ( ( 𝑌 ∈ 𝐾 ∧ 𝑆 ∈ 𝑌 ) → { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ ) |
118 |
113 117
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑆 ∈ 𝑌 ) → { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ ) |
119 |
111 118
|
anim12dan |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ ∧ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ ) ) |
120 |
|
r19.28zv |
⊢ ( { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ → ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
121 |
120
|
ralbidv |
⊢ ( { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ → ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
122 |
|
r19.27zv |
⊢ ( { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ → ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
123 |
121 122
|
sylan9bbr |
⊢ ( ( { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ≠ ∅ ∧ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ≠ ∅ ) → ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
124 |
119 123
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ( ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
125 |
104 124
|
bitrd |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
126 |
101
|
ralrab |
⊢ ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) |
127 |
95
|
ralrab |
⊢ ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ↔ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) |
128 |
126 127
|
anbi12i |
⊢ ( ( ∀ 𝑢 ∈ { 𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥 } ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ∧ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥 } ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ↔ ( ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) |
129 |
125 128
|
bitrdi |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( ∀ 𝑢 ∈ 𝐽 ∀ 𝑣 ∈ 𝐾 ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑢 × 𝑣 ) → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ ( 𝑢 × 𝑣 ) ) ↔ ( ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
130 |
17 129
|
syl5bb |
⊢ ( ( 𝜑 ∧ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) → ( ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ↔ ( ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
131 |
130
|
pm5.32da |
⊢ ( 𝜑 → ( ( ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ↔ ( ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ∧ ( ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) ) |
132 |
|
opelxp |
⊢ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ↔ ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ) |
133 |
132
|
anbi1i |
⊢ ( ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ↔ ( ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ) |
134 |
|
an4 |
⊢ ( ( ( 𝑅 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) ∧ ( 𝑆 ∈ 𝑌 ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ↔ ( ( 𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌 ) ∧ ( ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
135 |
131 133 134
|
3bitr4g |
⊢ ( 𝜑 → ( ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ↔ ( ( 𝑅 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) ∧ ( 𝑆 ∈ 𝑌 ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) ) |
136 |
|
eqid |
⊢ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) = ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) |
137 |
136
|
txval |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐽 ×t 𝐾 ) = ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) ) |
138 |
1 2 137
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ×t 𝐾 ) = ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) ) |
139 |
138
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐽 ×t 𝐾 ) fLimf 𝐿 ) = ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) fLimf 𝐿 ) ) |
140 |
139
|
fveq1d |
⊢ ( 𝜑 → ( ( ( 𝐽 ×t 𝐾 ) fLimf 𝐿 ) ‘ 𝐻 ) = ( ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) fLimf 𝐿 ) ‘ 𝐻 ) ) |
141 |
140
|
eleq2d |
⊢ ( 𝜑 → ( 〈 𝑅 , 𝑆 〉 ∈ ( ( ( 𝐽 ×t 𝐾 ) fLimf 𝐿 ) ‘ 𝐻 ) ↔ 〈 𝑅 , 𝑆 〉 ∈ ( ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) fLimf 𝐿 ) ‘ 𝐻 ) ) ) |
142 |
|
txtopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
143 |
1 2 142
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
144 |
138 143
|
eqeltrrd |
⊢ ( 𝜑 → ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
145 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) |
146 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑛 ) ∈ 𝑌 ) |
147 |
145 146
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 〈 ( 𝐹 ‘ 𝑛 ) , ( 𝐺 ‘ 𝑛 ) 〉 ∈ ( 𝑋 × 𝑌 ) ) |
148 |
147 6
|
fmptd |
⊢ ( 𝜑 → 𝐻 : 𝑍 ⟶ ( 𝑋 × 𝑌 ) ) |
149 |
|
eqid |
⊢ ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) = ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) |
150 |
149
|
flftg |
⊢ ( ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ∧ 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝐻 : 𝑍 ⟶ ( 𝑋 × 𝑌 ) ) → ( 〈 𝑅 , 𝑆 〉 ∈ ( ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) fLimf 𝐿 ) ‘ 𝐻 ) ↔ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ) ) |
151 |
144 3 148 150
|
syl3anc |
⊢ ( 𝜑 → ( 〈 𝑅 , 𝑆 〉 ∈ ( ( ( topGen ‘ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ) fLimf 𝐿 ) ‘ 𝐻 ) ↔ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ) ) |
152 |
141 151
|
bitrd |
⊢ ( 𝜑 → ( 〈 𝑅 , 𝑆 〉 ∈ ( ( ( 𝐽 ×t 𝐾 ) fLimf 𝐿 ) ‘ 𝐻 ) ↔ ( 〈 𝑅 , 𝑆 〉 ∈ ( 𝑋 × 𝑌 ) ∧ ∀ 𝑧 ∈ ran ( 𝑢 ∈ 𝐽 , 𝑣 ∈ 𝐾 ↦ ( 𝑢 × 𝑣 ) ) ( 〈 𝑅 , 𝑆 〉 ∈ 𝑧 → ∃ ℎ ∈ 𝐿 ( 𝐻 “ ℎ ) ⊆ 𝑧 ) ) ) ) |
153 |
|
isflf |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝐹 : 𝑍 ⟶ 𝑋 ) → ( 𝑅 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝑅 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) ) ) |
154 |
1 3 4 153
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝑅 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) ) ) |
155 |
|
isflf |
⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑍 ) ∧ 𝐺 : 𝑍 ⟶ 𝑌 ) → ( 𝑆 ∈ ( ( 𝐾 fLimf 𝐿 ) ‘ 𝐺 ) ↔ ( 𝑆 ∈ 𝑌 ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
156 |
2 3 5 155
|
syl3anc |
⊢ ( 𝜑 → ( 𝑆 ∈ ( ( 𝐾 fLimf 𝐿 ) ‘ 𝐺 ) ↔ ( 𝑆 ∈ 𝑌 ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) |
157 |
154 156
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑅 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ 𝑆 ∈ ( ( 𝐾 fLimf 𝐿 ) ‘ 𝐺 ) ) ↔ ( ( 𝑅 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑅 ∈ 𝑢 → ∃ 𝑓 ∈ 𝐿 ( 𝐹 “ 𝑓 ) ⊆ 𝑢 ) ) ∧ ( 𝑆 ∈ 𝑌 ∧ ∀ 𝑣 ∈ 𝐾 ( 𝑆 ∈ 𝑣 → ∃ 𝑔 ∈ 𝐿 ( 𝐺 “ 𝑔 ) ⊆ 𝑣 ) ) ) ) ) |
158 |
135 152 157
|
3bitr4d |
⊢ ( 𝜑 → ( 〈 𝑅 , 𝑆 〉 ∈ ( ( ( 𝐽 ×t 𝐾 ) fLimf 𝐿 ) ‘ 𝐻 ) ↔ ( 𝑅 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ 𝑆 ∈ ( ( 𝐾 fLimf 𝐿 ) ‘ 𝐺 ) ) ) ) |