Step |
Hyp |
Ref |
Expression |
1 |
|
mthmpps.r |
⊢ 𝑅 = ( mStRed ‘ 𝑇 ) |
2 |
|
mthmpps.j |
⊢ 𝐽 = ( mPPSt ‘ 𝑇 ) |
3 |
|
mthmpps.u |
⊢ 𝑈 = ( mThm ‘ 𝑇 ) |
4 |
|
mthmpps.d |
⊢ 𝐷 = ( mDV ‘ 𝑇 ) |
5 |
|
mthmpps.v |
⊢ 𝑉 = ( mVars ‘ 𝑇 ) |
6 |
|
mthmpps.z |
⊢ 𝑍 = ∪ ( 𝑉 “ ( 𝐻 ∪ { 𝐴 } ) ) |
7 |
|
mthmpps.m |
⊢ 𝑀 = ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
8 |
|
eqid |
⊢ ( mPreSt ‘ 𝑇 ) = ( mPreSt ‘ 𝑇 ) |
9 |
3 8
|
mthmsta |
⊢ 𝑈 ⊆ ( mPreSt ‘ 𝑇 ) |
10 |
|
simpr |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) |
11 |
9 10
|
sselid |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ) |
12 |
|
eqid |
⊢ ( mEx ‘ 𝑇 ) = ( mEx ‘ 𝑇 ) |
13 |
4 12 8
|
elmpst |
⊢ ( 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ↔ ( ( 𝐶 ⊆ 𝐷 ∧ ◡ 𝐶 = 𝐶 ) ∧ ( 𝐻 ⊆ ( mEx ‘ 𝑇 ) ∧ 𝐻 ∈ Fin ) ∧ 𝐴 ∈ ( mEx ‘ 𝑇 ) ) ) |
14 |
11 13
|
sylib |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( ( 𝐶 ⊆ 𝐷 ∧ ◡ 𝐶 = 𝐶 ) ∧ ( 𝐻 ⊆ ( mEx ‘ 𝑇 ) ∧ 𝐻 ∈ Fin ) ∧ 𝐴 ∈ ( mEx ‘ 𝑇 ) ) ) |
15 |
14
|
simp1d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝐶 ⊆ 𝐷 ∧ ◡ 𝐶 = 𝐶 ) ) |
16 |
15
|
simpld |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 𝐶 ⊆ 𝐷 ) |
17 |
|
difssd |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ⊆ 𝐷 ) |
18 |
16 17
|
unssd |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ⊆ 𝐷 ) |
19 |
7 18
|
eqsstrid |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 𝑀 ⊆ 𝐷 ) |
20 |
15
|
simprd |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ◡ 𝐶 = 𝐶 ) |
21 |
|
cnvdif |
⊢ ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) = ( ◡ 𝐷 ∖ ◡ ( 𝑍 × 𝑍 ) ) |
22 |
|
cnvdif |
⊢ ◡ ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) = ( ◡ ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ ◡ I ) |
23 |
|
cnvxp |
⊢ ◡ ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) = ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) |
24 |
|
cnvi |
⊢ ◡ I = I |
25 |
23 24
|
difeq12i |
⊢ ( ◡ ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ ◡ I ) = ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) |
26 |
22 25
|
eqtri |
⊢ ◡ ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) = ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) |
27 |
|
eqid |
⊢ ( mVR ‘ 𝑇 ) = ( mVR ‘ 𝑇 ) |
28 |
27 4
|
mdvval |
⊢ 𝐷 = ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) |
29 |
28
|
cnveqi |
⊢ ◡ 𝐷 = ◡ ( ( ( mVR ‘ 𝑇 ) × ( mVR ‘ 𝑇 ) ) ∖ I ) |
30 |
26 29 28
|
3eqtr4i |
⊢ ◡ 𝐷 = 𝐷 |
31 |
|
cnvxp |
⊢ ◡ ( 𝑍 × 𝑍 ) = ( 𝑍 × 𝑍 ) |
32 |
30 31
|
difeq12i |
⊢ ( ◡ 𝐷 ∖ ◡ ( 𝑍 × 𝑍 ) ) = ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) |
33 |
21 32
|
eqtri |
⊢ ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) = ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) |
34 |
33
|
a1i |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) = ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
35 |
20 34
|
uneq12d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( ◡ 𝐶 ∪ ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) = ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ) |
36 |
7
|
cnveqi |
⊢ ◡ 𝑀 = ◡ ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
37 |
|
cnvun |
⊢ ◡ ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) = ( ◡ 𝐶 ∪ ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
38 |
36 37
|
eqtri |
⊢ ◡ 𝑀 = ( ◡ 𝐶 ∪ ◡ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
39 |
35 38 7
|
3eqtr4g |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ◡ 𝑀 = 𝑀 ) |
40 |
19 39
|
jca |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝑀 ⊆ 𝐷 ∧ ◡ 𝑀 = 𝑀 ) ) |
41 |
14
|
simp2d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝐻 ⊆ ( mEx ‘ 𝑇 ) ∧ 𝐻 ∈ Fin ) ) |
42 |
14
|
simp3d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 𝐴 ∈ ( mEx ‘ 𝑇 ) ) |
43 |
4 12 8
|
elmpst |
⊢ ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ↔ ( ( 𝑀 ⊆ 𝐷 ∧ ◡ 𝑀 = 𝑀 ) ∧ ( 𝐻 ⊆ ( mEx ‘ 𝑇 ) ∧ 𝐻 ∈ Fin ) ∧ 𝐴 ∈ ( mEx ‘ 𝑇 ) ) ) |
44 |
40 41 42 43
|
syl3anbrc |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ) |
45 |
1 2 3
|
elmthm |
⊢ ( 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ↔ ∃ 𝑥 ∈ 𝐽 ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) |
46 |
10 45
|
sylib |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ∃ 𝑥 ∈ 𝐽 ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) |
47 |
|
eqid |
⊢ ( mCls ‘ 𝑇 ) = ( mCls ‘ 𝑇 ) |
48 |
|
simpll |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑇 ∈ mFS ) |
49 |
19
|
adantr |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑀 ⊆ 𝐷 ) |
50 |
41
|
simpld |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 𝐻 ⊆ ( mEx ‘ 𝑇 ) ) |
51 |
50
|
adantr |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝐻 ⊆ ( mEx ‘ 𝑇 ) ) |
52 |
8 2
|
mppspst |
⊢ 𝐽 ⊆ ( mPreSt ‘ 𝑇 ) |
53 |
|
simprl |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑥 ∈ 𝐽 ) |
54 |
52 53
|
sselid |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑥 ∈ ( mPreSt ‘ 𝑇 ) ) |
55 |
8
|
mpst123 |
⊢ ( 𝑥 ∈ ( mPreSt ‘ 𝑇 ) → 𝑥 = 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) |
56 |
54 55
|
syl |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑥 = 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) |
57 |
56
|
fveq2d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) ) |
58 |
|
simprr |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) |
59 |
57 58
|
eqtr3d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑅 ‘ 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) |
60 |
56 54
|
eqeltrrd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ∈ ( mPreSt ‘ 𝑇 ) ) |
61 |
|
eqid |
⊢ ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) = ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) |
62 |
5 8 1 61
|
msrval |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ∈ ( mPreSt ‘ 𝑇 ) → ( 𝑅 ‘ 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) = 〈 ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) |
63 |
60 62
|
syl |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑅 ‘ 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) = 〈 ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 ) |
64 |
5 8 1 6
|
msrval |
⊢ ( 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) → ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
65 |
11 64
|
syl |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
66 |
65
|
adantr |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
67 |
59 63 66
|
3eqtr3d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 〈 ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
68 |
|
fvex |
⊢ ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∈ V |
69 |
68
|
inex1 |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) ∈ V |
70 |
|
fvex |
⊢ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∈ V |
71 |
|
fvex |
⊢ ( 2nd ‘ 𝑥 ) ∈ V |
72 |
69 70 71
|
otth |
⊢ ( 〈 ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ↔ ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ∧ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) = 𝐻 ∧ ( 2nd ‘ 𝑥 ) = 𝐴 ) ) |
73 |
67 72
|
sylib |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ∧ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) = 𝐻 ∧ ( 2nd ‘ 𝑥 ) = 𝐴 ) ) |
74 |
73
|
simp1d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ) |
75 |
73
|
simp2d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 2nd ‘ ( 1st ‘ 𝑥 ) ) = 𝐻 ) |
76 |
73
|
simp3d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 2nd ‘ 𝑥 ) = 𝐴 ) |
77 |
76
|
sneqd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → { ( 2nd ‘ 𝑥 ) } = { 𝐴 } ) |
78 |
75 77
|
uneq12d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) = ( 𝐻 ∪ { 𝐴 } ) ) |
79 |
78
|
imaeq2d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑉 “ ( 𝐻 ∪ { 𝐴 } ) ) ) |
80 |
79
|
unieqd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) = ∪ ( 𝑉 “ ( 𝐻 ∪ { 𝐴 } ) ) ) |
81 |
80 6
|
eqtr4di |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) = 𝑍 ) |
82 |
81
|
sqxpeqd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) = ( 𝑍 × 𝑍 ) ) |
83 |
82
|
ineq2d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) × ∪ ( 𝑉 “ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ∪ { ( 2nd ‘ 𝑥 ) } ) ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ) |
84 |
74 83
|
eqtr3d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) = ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ) |
85 |
|
inss1 |
⊢ ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ⊆ 𝐶 |
86 |
84 85
|
eqsstrrdi |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ⊆ 𝐶 ) |
87 |
|
eqidd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑥 ) ) = ( 1st ‘ ( 1st ‘ 𝑥 ) ) ) |
88 |
87 75 76
|
oteq123d |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ 𝑥 ) 〉 = 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ) |
89 |
56 88
|
eqtrd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝑥 = 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ) |
90 |
89 54
|
eqeltrrd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ) |
91 |
4 12 8
|
elmpst |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ↔ ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ⊆ 𝐷 ∧ ◡ ( 1st ‘ ( 1st ‘ 𝑥 ) ) = ( 1st ‘ ( 1st ‘ 𝑥 ) ) ) ∧ ( 𝐻 ⊆ ( mEx ‘ 𝑇 ) ∧ 𝐻 ∈ Fin ) ∧ 𝐴 ∈ ( mEx ‘ 𝑇 ) ) ) |
92 |
91
|
simp1bi |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ⊆ 𝐷 ∧ ◡ ( 1st ‘ ( 1st ‘ 𝑥 ) ) = ( 1st ‘ ( 1st ‘ 𝑥 ) ) ) ) |
93 |
92
|
simpld |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) → ( 1st ‘ ( 1st ‘ 𝑥 ) ) ⊆ 𝐷 ) |
94 |
90 93
|
syl |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑥 ) ) ⊆ 𝐷 ) |
95 |
94
|
ssdifd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ⊆ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) |
96 |
|
unss12 |
⊢ ( ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ⊆ 𝐶 ∧ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ⊆ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ) ⊆ ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ) |
97 |
86 95 96
|
syl2anc |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ) ⊆ ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ) |
98 |
|
inundif |
⊢ ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ) = ( 1st ‘ ( 1st ‘ 𝑥 ) ) |
99 |
98
|
eqcomi |
⊢ ( 1st ‘ ( 1st ‘ 𝑥 ) ) = ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ∖ ( 𝑍 × 𝑍 ) ) ) |
100 |
97 99 7
|
3sstr4g |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑥 ) ) ⊆ 𝑀 ) |
101 |
|
ssidd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝐻 ⊆ 𝐻 ) |
102 |
4 12 47 48 49 51 100 101
|
ss2mcls |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( mCls ‘ 𝑇 ) 𝐻 ) ⊆ ( 𝑀 ( mCls ‘ 𝑇 ) 𝐻 ) ) |
103 |
89 53
|
eqeltrrd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ 𝐽 ) |
104 |
8 2 47
|
elmpps |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ 𝐽 ↔ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ∧ 𝐴 ∈ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( mCls ‘ 𝑇 ) 𝐻 ) ) ) |
105 |
104
|
simprbi |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , 𝐻 , 𝐴 〉 ∈ 𝐽 → 𝐴 ∈ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( mCls ‘ 𝑇 ) 𝐻 ) ) |
106 |
103 105
|
syl |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝐴 ∈ ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( mCls ‘ 𝑇 ) 𝐻 ) ) |
107 |
102 106
|
sseldd |
⊢ ( ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐽 ∧ ( 𝑅 ‘ 𝑥 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) → 𝐴 ∈ ( 𝑀 ( mCls ‘ 𝑇 ) 𝐻 ) ) |
108 |
46 107
|
rexlimddv |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 𝐴 ∈ ( 𝑀 ( mCls ‘ 𝑇 ) 𝐻 ) ) |
109 |
8 2 47
|
elmpps |
⊢ ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ↔ ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) ∧ 𝐴 ∈ ( 𝑀 ( mCls ‘ 𝑇 ) 𝐻 ) ) ) |
110 |
44 108 109
|
sylanbrc |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ) |
111 |
7
|
ineq1i |
⊢ ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) = ( ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ∩ ( 𝑍 × 𝑍 ) ) |
112 |
|
indir |
⊢ ( ( 𝐶 ∪ ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ) ∩ ( 𝑍 × 𝑍 ) ) = ( ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) ) |
113 |
|
disjdifr |
⊢ ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) = ∅ |
114 |
|
0ss |
⊢ ∅ ⊆ ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) |
115 |
113 114
|
eqsstri |
⊢ ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) ⊆ ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) |
116 |
|
ssequn2 |
⊢ ( ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) ⊆ ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ↔ ( ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ) |
117 |
115 116
|
mpbi |
⊢ ( ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ∪ ( ( 𝐷 ∖ ( 𝑍 × 𝑍 ) ) ∩ ( 𝑍 × 𝑍 ) ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) |
118 |
111 112 117
|
3eqtri |
⊢ ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) |
119 |
118
|
a1i |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) = ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) ) |
120 |
119
|
oteq1d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → 〈 ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 = 〈 ( 𝐶 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
121 |
5 8 1 6
|
msrval |
⊢ ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ ( mPreSt ‘ 𝑇 ) → ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = 〈 ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
122 |
44 121
|
syl |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = 〈 ( 𝑀 ∩ ( 𝑍 × 𝑍 ) ) , 𝐻 , 𝐴 〉 ) |
123 |
120 122 65
|
3eqtr4d |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) |
124 |
110 123
|
jca |
⊢ ( ( 𝑇 ∈ mFS ∧ 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) → ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ∧ ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) |
125 |
124
|
ex |
⊢ ( 𝑇 ∈ mFS → ( 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 → ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ∧ ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) ) |
126 |
1 2 3
|
mthmi |
⊢ ( ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ∧ ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) → 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ) |
127 |
125 126
|
impbid1 |
⊢ ( 𝑇 ∈ mFS → ( 〈 𝐶 , 𝐻 , 𝐴 〉 ∈ 𝑈 ↔ ( 〈 𝑀 , 𝐻 , 𝐴 〉 ∈ 𝐽 ∧ ( 𝑅 ‘ 〈 𝑀 , 𝐻 , 𝐴 〉 ) = ( 𝑅 ‘ 〈 𝐶 , 𝐻 , 𝐴 〉 ) ) ) ) |