Step |
Hyp |
Ref |
Expression |
1 |
|
pt1hmeo.j |
⊢ 𝐾 = ( ∏t ‘ { 〈 𝐴 , 𝐽 〉 } ) |
2 |
|
pt1hmeo.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
pt1hmeo.r |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
4 |
|
fconstmpt |
⊢ ( { 𝐴 } × { 𝑥 } ) = ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) |
5 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ 𝑉 ) |
6 |
|
sneq |
⊢ ( 𝑘 = 𝐴 → { 𝑘 } = { 𝐴 } ) |
7 |
6
|
xpeq1d |
⊢ ( 𝑘 = 𝐴 → ( { 𝑘 } × { 𝑥 } ) = ( { 𝐴 } × { 𝑥 } ) ) |
8 |
|
opeq1 |
⊢ ( 𝑘 = 𝐴 → 〈 𝑘 , 𝑥 〉 = 〈 𝐴 , 𝑥 〉 ) |
9 |
8
|
sneqd |
⊢ ( 𝑘 = 𝐴 → { 〈 𝑘 , 𝑥 〉 } = { 〈 𝐴 , 𝑥 〉 } ) |
10 |
7 9
|
eqeq12d |
⊢ ( 𝑘 = 𝐴 → ( ( { 𝑘 } × { 𝑥 } ) = { 〈 𝑘 , 𝑥 〉 } ↔ ( { 𝐴 } × { 𝑥 } ) = { 〈 𝐴 , 𝑥 〉 } ) ) |
11 |
|
vex |
⊢ 𝑘 ∈ V |
12 |
|
vex |
⊢ 𝑥 ∈ V |
13 |
11 12
|
xpsn |
⊢ ( { 𝑘 } × { 𝑥 } ) = { 〈 𝑘 , 𝑥 〉 } |
14 |
10 13
|
vtoclg |
⊢ ( 𝐴 ∈ 𝑉 → ( { 𝐴 } × { 𝑥 } ) = { 〈 𝐴 , 𝑥 〉 } ) |
15 |
5 14
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( { 𝐴 } × { 𝑥 } ) = { 〈 𝐴 , 𝑥 〉 } ) |
16 |
4 15
|
syl5eqr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) = { 〈 𝐴 , 𝑥 〉 } ) |
17 |
16
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ) = ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ) |
18 |
|
snex |
⊢ { 𝐴 } ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → { 𝐴 } ∈ V ) |
20 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
21 |
3 20
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
22 |
2 21
|
fsnd |
⊢ ( 𝜑 → { 〈 𝐴 , 𝐽 〉 } : { 𝐴 } ⟶ Top ) |
23 |
3
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) ) |
25 |
|
elsni |
⊢ ( 𝑘 ∈ { 𝐴 } → 𝑘 = 𝐴 ) |
26 |
25
|
fveq2d |
⊢ ( 𝑘 ∈ { 𝐴 } → ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝑘 ) = ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) ) |
27 |
|
fvsng |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) → ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) = 𝐽 ) |
28 |
2 3 27
|
syl2anc |
⊢ ( 𝜑 → ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) = 𝐽 ) |
29 |
26 28
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝑘 ) = 𝐽 ) |
30 |
29
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → ( 𝐽 Cn ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝑘 ) ) = ( 𝐽 Cn 𝐽 ) ) |
31 |
24 30
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → ( 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( 𝐽 Cn ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝑘 ) ) ) |
32 |
1 3 19 22 31
|
ptcn |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ) ∈ ( 𝐽 Cn 𝐾 ) ) |
33 |
17 32
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐽 Cn 𝐾 ) ) |
34 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) |
35 |
16
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) = { 〈 𝐴 , 𝑥 〉 } ) |
36 |
34 35
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑦 = ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ) |
37 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑥 ∈ 𝑋 ) |
38 |
37
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) ∧ 𝑘 ∈ { 𝐴 } ) → 𝑥 ∈ 𝑋 ) |
39 |
38
|
fmpttd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) : { 𝐴 } ⟶ 𝑋 ) |
40 |
|
toponmax |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 ∈ 𝐽 ) |
41 |
3 40
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑋 ∈ 𝐽 ) |
43 |
|
elmapg |
⊢ ( ( 𝑋 ∈ 𝐽 ∧ { 𝐴 } ∈ V ) → ( ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ∈ ( 𝑋 ↑m { 𝐴 } ) ↔ ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) : { 𝐴 } ⟶ 𝑋 ) ) |
44 |
42 18 43
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ∈ ( 𝑋 ↑m { 𝐴 } ) ↔ ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) : { 𝐴 } ⟶ 𝑋 ) ) |
45 |
39 44
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( 𝑘 ∈ { 𝐴 } ↦ 𝑥 ) ∈ ( 𝑋 ↑m { 𝐴 } ) ) |
46 |
36 45
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ) |
47 |
34
|
fveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( 𝑦 ‘ 𝐴 ) = ( { 〈 𝐴 , 𝑥 〉 } ‘ 𝐴 ) ) |
48 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝐴 ∈ 𝑉 ) |
49 |
|
fvsng |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋 ) → ( { 〈 𝐴 , 𝑥 〉 } ‘ 𝐴 ) = 𝑥 ) |
50 |
48 37 49
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( { 〈 𝐴 , 𝑥 〉 } ‘ 𝐴 ) = 𝑥 ) |
51 |
47 50
|
eqtr2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → 𝑥 = ( 𝑦 ‘ 𝐴 ) ) |
52 |
46 51
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) → ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) |
53 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑥 = ( 𝑦 ‘ 𝐴 ) ) |
54 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ) |
55 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑋 ∈ 𝐽 ) |
56 |
|
elmapg |
⊢ ( ( 𝑋 ∈ 𝐽 ∧ { 𝐴 } ∈ V ) → ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ↔ 𝑦 : { 𝐴 } ⟶ 𝑋 ) ) |
57 |
55 18 56
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ↔ 𝑦 : { 𝐴 } ⟶ 𝑋 ) ) |
58 |
54 57
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑦 : { 𝐴 } ⟶ 𝑋 ) |
59 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
60 |
2 59
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ { 𝐴 } ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝐴 ∈ { 𝐴 } ) |
62 |
58 61
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → ( 𝑦 ‘ 𝐴 ) ∈ 𝑋 ) |
63 |
53 62
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑥 ∈ 𝑋 ) |
64 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝐴 ∈ 𝑉 ) |
65 |
|
fsn2g |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 : { 𝐴 } ⟶ 𝑋 ↔ ( ( 𝑦 ‘ 𝐴 ) ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 } ) ) ) |
66 |
64 65
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → ( 𝑦 : { 𝐴 } ⟶ 𝑋 ↔ ( ( 𝑦 ‘ 𝐴 ) ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 } ) ) ) |
67 |
58 66
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → ( ( 𝑦 ‘ 𝐴 ) ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 } ) ) |
68 |
67
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑦 = { 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 } ) |
69 |
53
|
opeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 〈 𝐴 , 𝑥 〉 = 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 ) |
70 |
69
|
sneqd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → { 〈 𝐴 , 𝑥 〉 } = { 〈 𝐴 , ( 𝑦 ‘ 𝐴 ) 〉 } ) |
71 |
68 70
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) |
72 |
63 71
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) → ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ) |
73 |
52 72
|
impbida |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝑋 ∧ 𝑦 = { 〈 𝐴 , 𝑥 〉 } ) ↔ ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ∧ 𝑥 = ( 𝑦 ‘ 𝐴 ) ) ) ) |
74 |
73
|
mptcnv |
⊢ ( 𝜑 → ◡ ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) = ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ↦ ( 𝑦 ‘ 𝐴 ) ) ) |
75 |
|
xpsng |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) → ( { 𝐴 } × { 𝐽 } ) = { 〈 𝐴 , 𝐽 〉 } ) |
76 |
2 3 75
|
syl2anc |
⊢ ( 𝜑 → ( { 𝐴 } × { 𝐽 } ) = { 〈 𝐴 , 𝐽 〉 } ) |
77 |
76
|
eqcomd |
⊢ ( 𝜑 → { 〈 𝐴 , 𝐽 〉 } = ( { 𝐴 } × { 𝐽 } ) ) |
78 |
77
|
fveq2d |
⊢ ( 𝜑 → ( ∏t ‘ { 〈 𝐴 , 𝐽 〉 } ) = ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) ) |
79 |
1 78
|
syl5eq |
⊢ ( 𝜑 → 𝐾 = ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) ) |
80 |
|
eqid |
⊢ ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) = ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) |
81 |
80
|
pttoponconst |
⊢ ( ( { 𝐴 } ∈ V ∧ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) → ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) ∈ ( TopOn ‘ ( 𝑋 ↑m { 𝐴 } ) ) ) |
82 |
19 3 81
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ ( { 𝐴 } × { 𝐽 } ) ) ∈ ( TopOn ‘ ( 𝑋 ↑m { 𝐴 } ) ) ) |
83 |
79 82
|
eqeltrd |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ ( 𝑋 ↑m { 𝐴 } ) ) ) |
84 |
|
toponuni |
⊢ ( 𝐾 ∈ ( TopOn ‘ ( 𝑋 ↑m { 𝐴 } ) ) → ( 𝑋 ↑m { 𝐴 } ) = ∪ 𝐾 ) |
85 |
83 84
|
syl |
⊢ ( 𝜑 → ( 𝑋 ↑m { 𝐴 } ) = ∪ 𝐾 ) |
86 |
85
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝑋 ↑m { 𝐴 } ) ↦ ( 𝑦 ‘ 𝐴 ) ) = ( 𝑦 ∈ ∪ 𝐾 ↦ ( 𝑦 ‘ 𝐴 ) ) ) |
87 |
74 86
|
eqtrd |
⊢ ( 𝜑 → ◡ ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) = ( 𝑦 ∈ ∪ 𝐾 ↦ ( 𝑦 ‘ 𝐴 ) ) ) |
88 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
89 |
88 1
|
ptpjcn |
⊢ ( ( { 𝐴 } ∈ V ∧ { 〈 𝐴 , 𝐽 〉 } : { 𝐴 } ⟶ Top ∧ 𝐴 ∈ { 𝐴 } ) → ( 𝑦 ∈ ∪ 𝐾 ↦ ( 𝑦 ‘ 𝐴 ) ) ∈ ( 𝐾 Cn ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) ) ) |
90 |
18 22 60 89
|
mp3an2i |
⊢ ( 𝜑 → ( 𝑦 ∈ ∪ 𝐾 ↦ ( 𝑦 ‘ 𝐴 ) ) ∈ ( 𝐾 Cn ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) ) ) |
91 |
28
|
oveq2d |
⊢ ( 𝜑 → ( 𝐾 Cn ( { 〈 𝐴 , 𝐽 〉 } ‘ 𝐴 ) ) = ( 𝐾 Cn 𝐽 ) ) |
92 |
90 91
|
eleqtrd |
⊢ ( 𝜑 → ( 𝑦 ∈ ∪ 𝐾 ↦ ( 𝑦 ‘ 𝐴 ) ) ∈ ( 𝐾 Cn 𝐽 ) ) |
93 |
87 92
|
eqeltrd |
⊢ ( 𝜑 → ◡ ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐾 Cn 𝐽 ) ) |
94 |
|
ishmeo |
⊢ ( ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐽 Homeo 𝐾 ) ↔ ( ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐽 Cn 𝐾 ) ∧ ◡ ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐾 Cn 𝐽 ) ) ) |
95 |
33 93 94
|
sylanbrc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ { 〈 𝐴 , 𝑥 〉 } ) ∈ ( 𝐽 Homeo 𝐾 ) ) |