Step |
Hyp |
Ref |
Expression |
1 |
|
isssc.1 |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
2 |
|
isssc.2 |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑇 × 𝑇 ) ) |
3 |
|
isssc.3 |
⊢ ( 𝜑 → 𝑇 ∈ 𝑉 ) |
4 |
|
brssc |
⊢ ( 𝐻 ⊆cat 𝐽 ↔ ∃ 𝑡 ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
5 |
|
fndm |
⊢ ( 𝐽 Fn ( 𝑡 × 𝑡 ) → dom 𝐽 = ( 𝑡 × 𝑡 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → dom 𝐽 = ( 𝑡 × 𝑡 ) ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → 𝐽 Fn ( 𝑇 × 𝑇 ) ) |
8 |
7
|
fndmd |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → dom 𝐽 = ( 𝑇 × 𝑇 ) ) |
9 |
6 8
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → ( 𝑡 × 𝑡 ) = ( 𝑇 × 𝑇 ) ) |
10 |
9
|
dmeqd |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → dom ( 𝑡 × 𝑡 ) = dom ( 𝑇 × 𝑇 ) ) |
11 |
|
dmxpid |
⊢ dom ( 𝑡 × 𝑡 ) = 𝑡 |
12 |
|
dmxpid |
⊢ dom ( 𝑇 × 𝑇 ) = 𝑇 |
13 |
10 11 12
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ 𝐽 Fn ( 𝑡 × 𝑡 ) ) → 𝑡 = 𝑇 ) |
14 |
13
|
ex |
⊢ ( 𝜑 → ( 𝐽 Fn ( 𝑡 × 𝑡 ) → 𝑡 = 𝑇 ) ) |
15 |
|
id |
⊢ ( 𝑡 = 𝑇 → 𝑡 = 𝑇 ) |
16 |
15
|
sqxpeqd |
⊢ ( 𝑡 = 𝑇 → ( 𝑡 × 𝑡 ) = ( 𝑇 × 𝑇 ) ) |
17 |
16
|
fneq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝐽 Fn ( 𝑡 × 𝑡 ) ↔ 𝐽 Fn ( 𝑇 × 𝑇 ) ) ) |
18 |
2 17
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑡 = 𝑇 → 𝐽 Fn ( 𝑡 × 𝑡 ) ) ) |
19 |
14 18
|
impbid |
⊢ ( 𝜑 → ( 𝐽 Fn ( 𝑡 × 𝑡 ) ↔ 𝑡 = 𝑇 ) ) |
20 |
19
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝑡 = 𝑇 ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
21 |
20
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑡 ( 𝐽 Fn ( 𝑡 × 𝑡 ) ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ∃ 𝑡 ( 𝑡 = 𝑇 ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
22 |
4 21
|
syl5bb |
⊢ ( 𝜑 → ( 𝐻 ⊆cat 𝐽 ↔ ∃ 𝑡 ( 𝑡 = 𝑇 ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
23 |
|
pweq |
⊢ ( 𝑡 = 𝑇 → 𝒫 𝑡 = 𝒫 𝑇 ) |
24 |
23
|
rexeqdv |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
25 |
24
|
ceqsexgv |
⊢ ( 𝑇 ∈ 𝑉 → ( ∃ 𝑡 ( 𝑡 = 𝑇 ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
26 |
3 25
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑡 ( 𝑡 = 𝑇 ∧ ∃ 𝑠 ∈ 𝒫 𝑡 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
27 |
22 26
|
bitrd |
⊢ ( 𝜑 → ( 𝐻 ⊆cat 𝐽 ↔ ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
28 |
|
df-rex |
⊢ ( ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∃ 𝑠 ( 𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
29 |
|
3anass |
⊢ ( ( 𝐻 ∈ V ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝐻 ∈ V ∧ ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
30 |
|
elixp2 |
⊢ ( 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ( 𝐻 ∈ V ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
31 |
|
vex |
⊢ 𝑠 ∈ V |
32 |
31 31
|
xpex |
⊢ ( 𝑠 × 𝑠 ) ∈ V |
33 |
|
fnex |
⊢ ( ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ( 𝑠 × 𝑠 ) ∈ V ) → 𝐻 ∈ V ) |
34 |
32 33
|
mpan2 |
⊢ ( 𝐻 Fn ( 𝑠 × 𝑠 ) → 𝐻 ∈ V ) |
35 |
34
|
adantr |
⊢ ( ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) → 𝐻 ∈ V ) |
36 |
35
|
pm4.71ri |
⊢ ( ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝐻 ∈ V ∧ ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
37 |
29 30 36
|
3bitr4i |
⊢ ( 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
38 |
|
fndm |
⊢ ( 𝐻 Fn ( 𝑠 × 𝑠 ) → dom 𝐻 = ( 𝑠 × 𝑠 ) ) |
39 |
38
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom 𝐻 = ( 𝑠 × 𝑠 ) ) |
40 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
41 |
40
|
fndmd |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom 𝐻 = ( 𝑆 × 𝑆 ) ) |
42 |
39 41
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → ( 𝑠 × 𝑠 ) = ( 𝑆 × 𝑆 ) ) |
43 |
42
|
dmeqd |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → dom ( 𝑠 × 𝑠 ) = dom ( 𝑆 × 𝑆 ) ) |
44 |
|
dmxpid |
⊢ dom ( 𝑠 × 𝑠 ) = 𝑠 |
45 |
|
dmxpid |
⊢ dom ( 𝑆 × 𝑆 ) = 𝑆 |
46 |
43 44 45
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ 𝐻 Fn ( 𝑠 × 𝑠 ) ) → 𝑠 = 𝑆 ) |
47 |
46
|
ex |
⊢ ( 𝜑 → ( 𝐻 Fn ( 𝑠 × 𝑠 ) → 𝑠 = 𝑆 ) ) |
48 |
|
id |
⊢ ( 𝑠 = 𝑆 → 𝑠 = 𝑆 ) |
49 |
48
|
sqxpeqd |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 × 𝑠 ) = ( 𝑆 × 𝑆 ) ) |
50 |
49
|
fneq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝐻 Fn ( 𝑠 × 𝑠 ) ↔ 𝐻 Fn ( 𝑆 × 𝑆 ) ) ) |
51 |
1 50
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑠 = 𝑆 → 𝐻 Fn ( 𝑠 × 𝑠 ) ) ) |
52 |
47 51
|
impbid |
⊢ ( 𝜑 → ( 𝐻 Fn ( 𝑠 × 𝑠 ) ↔ 𝑠 = 𝑆 ) ) |
53 |
52
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝐻 Fn ( 𝑠 × 𝑠 ) ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝑠 = 𝑆 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
54 |
37 53
|
syl5bb |
⊢ ( 𝜑 → ( 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ( 𝑠 = 𝑆 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
55 |
54
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝑠 ∈ 𝒫 𝑇 ∧ ( 𝑠 = 𝑆 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) ) |
56 |
|
an12 |
⊢ ( ( 𝑠 ∈ 𝒫 𝑇 ∧ ( 𝑠 = 𝑆 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ↔ ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) |
57 |
55 56
|
bitrdi |
⊢ ( 𝜑 → ( ( 𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) ) |
58 |
57
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑠 ( 𝑠 ∈ 𝒫 𝑇 ∧ 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) ) |
59 |
28 58
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝒫 𝑇 𝐻 ∈ X 𝑧 ∈ ( 𝑠 × 𝑠 ) 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ) ) |
60 |
|
exsimpl |
⊢ ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) → ∃ 𝑠 𝑠 = 𝑆 ) |
61 |
|
isset |
⊢ ( 𝑆 ∈ V ↔ ∃ 𝑠 𝑠 = 𝑆 ) |
62 |
60 61
|
sylibr |
⊢ ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) → 𝑆 ∈ V ) |
63 |
62
|
a1i |
⊢ ( 𝜑 → ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) → 𝑆 ∈ V ) ) |
64 |
|
ssexg |
⊢ ( ( 𝑆 ⊆ 𝑇 ∧ 𝑇 ∈ 𝑉 ) → 𝑆 ∈ V ) |
65 |
64
|
expcom |
⊢ ( 𝑇 ∈ 𝑉 → ( 𝑆 ⊆ 𝑇 → 𝑆 ∈ V ) ) |
66 |
3 65
|
syl |
⊢ ( 𝜑 → ( 𝑆 ⊆ 𝑇 → 𝑆 ∈ V ) ) |
67 |
66
|
adantrd |
⊢ ( 𝜑 → ( ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) → 𝑆 ∈ V ) ) |
68 |
31
|
elpw |
⊢ ( 𝑠 ∈ 𝒫 𝑇 ↔ 𝑠 ⊆ 𝑇 ) |
69 |
|
sseq1 |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇 ) ) |
70 |
68 69
|
syl5bb |
⊢ ( 𝑠 = 𝑆 → ( 𝑠 ∈ 𝒫 𝑇 ↔ 𝑆 ⊆ 𝑇 ) ) |
71 |
49
|
raleqdv |
⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∀ 𝑧 ∈ ( 𝑆 × 𝑆 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) |
72 |
|
fvex |
⊢ ( 𝐻 ‘ 𝑧 ) ∈ V |
73 |
72
|
elpw |
⊢ ( ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ( 𝐻 ‘ 𝑧 ) ⊆ ( 𝐽 ‘ 𝑧 ) ) |
74 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
75 |
|
df-ov |
⊢ ( 𝑥 𝐻 𝑦 ) = ( 𝐻 ‘ 〈 𝑥 , 𝑦 〉 ) |
76 |
74 75
|
eqtr4di |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐻 ‘ 𝑧 ) = ( 𝑥 𝐻 𝑦 ) ) |
77 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐽 ‘ 𝑧 ) = ( 𝐽 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
78 |
|
df-ov |
⊢ ( 𝑥 𝐽 𝑦 ) = ( 𝐽 ‘ 〈 𝑥 , 𝑦 〉 ) |
79 |
77 78
|
eqtr4di |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐽 ‘ 𝑧 ) = ( 𝑥 𝐽 𝑦 ) ) |
80 |
76 79
|
sseq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐻 ‘ 𝑧 ) ⊆ ( 𝐽 ‘ 𝑧 ) ↔ ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) |
81 |
73 80
|
syl5bb |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) |
82 |
81
|
ralxp |
⊢ ( ∀ 𝑧 ∈ ( 𝑆 × 𝑆 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) |
83 |
71 82
|
bitrdi |
⊢ ( 𝑠 = 𝑆 → ( ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) |
84 |
70 83
|
anbi12d |
⊢ ( 𝑠 = 𝑆 → ( ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) |
85 |
84
|
ceqsexgv |
⊢ ( 𝑆 ∈ V → ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) |
86 |
85
|
a1i |
⊢ ( 𝜑 → ( 𝑆 ∈ V → ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) ) |
87 |
63 67 86
|
pm5.21ndd |
⊢ ( 𝜑 → ( ∃ 𝑠 ( 𝑠 = 𝑆 ∧ ( 𝑠 ∈ 𝒫 𝑇 ∧ ∀ 𝑧 ∈ ( 𝑠 × 𝑠 ) ( 𝐻 ‘ 𝑧 ) ∈ 𝒫 ( 𝐽 ‘ 𝑧 ) ) ) ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) |
88 |
27 59 87
|
3bitrd |
⊢ ( 𝜑 → ( 𝐻 ⊆cat 𝐽 ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) |