Step |
Hyp |
Ref |
Expression |
1 |
|
ntrcls.o |
|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) ) |
2 |
|
ntrcls.d |
|- D = ( O ` B ) |
3 |
|
ntrcls.r |
|- ( ph -> I D K ) |
4 |
|
ineq1 |
|- ( s = a -> ( s i^i t ) = ( a i^i t ) ) |
5 |
4
|
fveq2d |
|- ( s = a -> ( I ` ( s i^i t ) ) = ( I ` ( a i^i t ) ) ) |
6 |
|
fveq2 |
|- ( s = a -> ( I ` s ) = ( I ` a ) ) |
7 |
6
|
ineq1d |
|- ( s = a -> ( ( I ` s ) i^i ( I ` t ) ) = ( ( I ` a ) i^i ( I ` t ) ) ) |
8 |
5 7
|
eqeq12d |
|- ( s = a -> ( ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> ( I ` ( a i^i t ) ) = ( ( I ` a ) i^i ( I ` t ) ) ) ) |
9 |
|
ineq2 |
|- ( t = b -> ( a i^i t ) = ( a i^i b ) ) |
10 |
9
|
fveq2d |
|- ( t = b -> ( I ` ( a i^i t ) ) = ( I ` ( a i^i b ) ) ) |
11 |
|
fveq2 |
|- ( t = b -> ( I ` t ) = ( I ` b ) ) |
12 |
11
|
ineq2d |
|- ( t = b -> ( ( I ` a ) i^i ( I ` t ) ) = ( ( I ` a ) i^i ( I ` b ) ) ) |
13 |
10 12
|
eqeq12d |
|- ( t = b -> ( ( I ` ( a i^i t ) ) = ( ( I ` a ) i^i ( I ` t ) ) <-> ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) ) ) |
14 |
8 13
|
cbvral2vw |
|- ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> A. a e. ~P B A. b e. ~P B ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) ) |
15 |
2 3
|
ntrclsbex |
|- ( ph -> B e. _V ) |
16 |
|
difssd |
|- ( ph -> ( B \ s ) C_ B ) |
17 |
15 16
|
sselpwd |
|- ( ph -> ( B \ s ) e. ~P B ) |
18 |
17
|
adantr |
|- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B ) |
19 |
|
elpwi |
|- ( a e. ~P B -> a C_ B ) |
20 |
15
|
adantr |
|- ( ( ph /\ a C_ B ) -> B e. _V ) |
21 |
|
difssd |
|- ( ( ph /\ a C_ B ) -> ( B \ a ) C_ B ) |
22 |
20 21
|
sselpwd |
|- ( ( ph /\ a C_ B ) -> ( B \ a ) e. ~P B ) |
23 |
|
difeq2 |
|- ( s = ( B \ a ) -> ( B \ s ) = ( B \ ( B \ a ) ) ) |
24 |
23
|
eqeq2d |
|- ( s = ( B \ a ) -> ( a = ( B \ s ) <-> a = ( B \ ( B \ a ) ) ) ) |
25 |
|
eqcom |
|- ( a = ( B \ ( B \ a ) ) <-> ( B \ ( B \ a ) ) = a ) |
26 |
24 25
|
bitrdi |
|- ( s = ( B \ a ) -> ( a = ( B \ s ) <-> ( B \ ( B \ a ) ) = a ) ) |
27 |
26
|
adantl |
|- ( ( ( ph /\ a C_ B ) /\ s = ( B \ a ) ) -> ( a = ( B \ s ) <-> ( B \ ( B \ a ) ) = a ) ) |
28 |
|
dfss4 |
|- ( a C_ B <-> ( B \ ( B \ a ) ) = a ) |
29 |
28
|
biimpi |
|- ( a C_ B -> ( B \ ( B \ a ) ) = a ) |
30 |
29
|
adantl |
|- ( ( ph /\ a C_ B ) -> ( B \ ( B \ a ) ) = a ) |
31 |
22 27 30
|
rspcedvd |
|- ( ( ph /\ a C_ B ) -> E. s e. ~P B a = ( B \ s ) ) |
32 |
19 31
|
sylan2 |
|- ( ( ph /\ a e. ~P B ) -> E. s e. ~P B a = ( B \ s ) ) |
33 |
|
ineq1 |
|- ( a = ( B \ s ) -> ( a i^i b ) = ( ( B \ s ) i^i b ) ) |
34 |
33
|
fveq2d |
|- ( a = ( B \ s ) -> ( I ` ( a i^i b ) ) = ( I ` ( ( B \ s ) i^i b ) ) ) |
35 |
|
fveq2 |
|- ( a = ( B \ s ) -> ( I ` a ) = ( I ` ( B \ s ) ) ) |
36 |
35
|
ineq1d |
|- ( a = ( B \ s ) -> ( ( I ` a ) i^i ( I ` b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) ) |
37 |
34 36
|
eqeq12d |
|- ( a = ( B \ s ) -> ( ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) <-> ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) ) ) |
38 |
37
|
ralbidv |
|- ( a = ( B \ s ) -> ( A. b e. ~P B ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) <-> A. b e. ~P B ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) ) ) |
39 |
38
|
3ad2ant3 |
|- ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) -> ( A. b e. ~P B ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) <-> A. b e. ~P B ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) ) ) |
40 |
|
difssd |
|- ( ph -> ( B \ t ) C_ B ) |
41 |
15 40
|
sselpwd |
|- ( ph -> ( B \ t ) e. ~P B ) |
42 |
41
|
ad2antrr |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B ) -> ( B \ t ) e. ~P B ) |
43 |
|
simpll |
|- ( ( ( ph /\ s e. ~P B ) /\ b e. ~P B ) -> ph ) |
44 |
|
elpwi |
|- ( b e. ~P B -> b C_ B ) |
45 |
44
|
adantl |
|- ( ( ( ph /\ s e. ~P B ) /\ b e. ~P B ) -> b C_ B ) |
46 |
|
difssd |
|- ( ph -> ( B \ b ) C_ B ) |
47 |
15 46
|
sselpwd |
|- ( ph -> ( B \ b ) e. ~P B ) |
48 |
47
|
adantr |
|- ( ( ph /\ b C_ B ) -> ( B \ b ) e. ~P B ) |
49 |
|
difeq2 |
|- ( t = ( B \ b ) -> ( B \ t ) = ( B \ ( B \ b ) ) ) |
50 |
49
|
eqeq2d |
|- ( t = ( B \ b ) -> ( b = ( B \ t ) <-> b = ( B \ ( B \ b ) ) ) ) |
51 |
|
eqcom |
|- ( b = ( B \ ( B \ b ) ) <-> ( B \ ( B \ b ) ) = b ) |
52 |
50 51
|
bitrdi |
|- ( t = ( B \ b ) -> ( b = ( B \ t ) <-> ( B \ ( B \ b ) ) = b ) ) |
53 |
52
|
adantl |
|- ( ( ( ph /\ b C_ B ) /\ t = ( B \ b ) ) -> ( b = ( B \ t ) <-> ( B \ ( B \ b ) ) = b ) ) |
54 |
|
dfss4 |
|- ( b C_ B <-> ( B \ ( B \ b ) ) = b ) |
55 |
54
|
biimpi |
|- ( b C_ B -> ( B \ ( B \ b ) ) = b ) |
56 |
55
|
adantl |
|- ( ( ph /\ b C_ B ) -> ( B \ ( B \ b ) ) = b ) |
57 |
48 53 56
|
rspcedvd |
|- ( ( ph /\ b C_ B ) -> E. t e. ~P B b = ( B \ t ) ) |
58 |
43 45 57
|
syl2anc |
|- ( ( ( ph /\ s e. ~P B ) /\ b e. ~P B ) -> E. t e. ~P B b = ( B \ t ) ) |
59 |
|
ineq2 |
|- ( b = ( B \ t ) -> ( ( B \ s ) i^i b ) = ( ( B \ s ) i^i ( B \ t ) ) ) |
60 |
|
difundi |
|- ( B \ ( s u. t ) ) = ( ( B \ s ) i^i ( B \ t ) ) |
61 |
59 60
|
eqtr4di |
|- ( b = ( B \ t ) -> ( ( B \ s ) i^i b ) = ( B \ ( s u. t ) ) ) |
62 |
61
|
fveq2d |
|- ( b = ( B \ t ) -> ( I ` ( ( B \ s ) i^i b ) ) = ( I ` ( B \ ( s u. t ) ) ) ) |
63 |
|
fveq2 |
|- ( b = ( B \ t ) -> ( I ` b ) = ( I ` ( B \ t ) ) ) |
64 |
63
|
ineq2d |
|- ( b = ( B \ t ) -> ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) |
65 |
62 64
|
eqeq12d |
|- ( b = ( B \ t ) -> ( ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) <-> ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) |
66 |
65
|
3ad2ant3 |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) <-> ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) |
67 |
|
simp1l |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ph ) |
68 |
67 15
|
jccir |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ph /\ B e. _V ) ) |
69 |
|
simp1r |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> s e. ~P B ) |
70 |
|
simp2 |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> t e. ~P B ) |
71 |
1 2 3
|
ntrclsiex |
|- ( ph -> I e. ( ~P B ^m ~P B ) ) |
72 |
|
elmapi |
|- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B ) |
73 |
71 72
|
syl |
|- ( ph -> I : ~P B --> ~P B ) |
74 |
73
|
anim1i |
|- ( ( ph /\ B e. _V ) -> ( I : ~P B --> ~P B /\ B e. _V ) ) |
75 |
74
|
adantr |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( I : ~P B --> ~P B /\ B e. _V ) ) |
76 |
|
simpl |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> I : ~P B --> ~P B ) |
77 |
|
simpr |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> B e. _V ) |
78 |
|
difssd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( B \ ( s u. t ) ) C_ B ) |
79 |
77 78
|
sselpwd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( B \ ( s u. t ) ) e. ~P B ) |
80 |
76 79
|
ffvelrnd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( I ` ( B \ ( s u. t ) ) ) e. ~P B ) |
81 |
80
|
elpwid |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( I ` ( B \ ( s u. t ) ) ) C_ B ) |
82 |
|
difssd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( B \ s ) C_ B ) |
83 |
77 82
|
sselpwd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( B \ s ) e. ~P B ) |
84 |
76 83
|
ffvelrnd |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( I ` ( B \ s ) ) e. ~P B ) |
85 |
84
|
elpwid |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( I ` ( B \ s ) ) C_ B ) |
86 |
|
ssinss1 |
|- ( ( I ` ( B \ s ) ) C_ B -> ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) C_ B ) |
87 |
85 86
|
syl |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) C_ B ) |
88 |
81 87
|
jca |
|- ( ( I : ~P B --> ~P B /\ B e. _V ) -> ( ( I ` ( B \ ( s u. t ) ) ) C_ B /\ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) C_ B ) ) |
89 |
|
rcompleq |
|- ( ( ( I ` ( B \ ( s u. t ) ) ) C_ B /\ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) C_ B ) -> ( ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) <-> ( B \ ( I ` ( B \ ( s u. t ) ) ) ) = ( B \ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) ) |
90 |
75 88 89
|
3syl |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) <-> ( B \ ( I ` ( B \ ( s u. t ) ) ) ) = ( B \ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) ) |
91 |
|
simplr |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> B e. _V ) |
92 |
71
|
ad2antrr |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> I e. ( ~P B ^m ~P B ) ) |
93 |
|
eqid |
|- ( D ` I ) = ( D ` I ) |
94 |
|
simprl |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> s e. ~P B ) |
95 |
94
|
elpwid |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> s C_ B ) |
96 |
|
simprr |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> t e. ~P B ) |
97 |
96
|
elpwid |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> t C_ B ) |
98 |
95 97
|
unssd |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( s u. t ) C_ B ) |
99 |
91 98
|
sselpwd |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( s u. t ) e. ~P B ) |
100 |
|
eqid |
|- ( ( D ` I ) ` ( s u. t ) ) = ( ( D ` I ) ` ( s u. t ) ) |
101 |
1 2 91 92 93 99 100
|
dssmapfv3d |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( D ` I ) ` ( s u. t ) ) = ( B \ ( I ` ( B \ ( s u. t ) ) ) ) ) |
102 |
|
simpl |
|- ( ( s e. ~P B /\ t e. ~P B ) -> s e. ~P B ) |
103 |
|
simplr |
|- ( ( ( ph /\ B e. _V ) /\ s e. ~P B ) -> B e. _V ) |
104 |
71
|
ad2antrr |
|- ( ( ( ph /\ B e. _V ) /\ s e. ~P B ) -> I e. ( ~P B ^m ~P B ) ) |
105 |
|
simpr |
|- ( ( ( ph /\ B e. _V ) /\ s e. ~P B ) -> s e. ~P B ) |
106 |
|
eqid |
|- ( ( D ` I ) ` s ) = ( ( D ` I ) ` s ) |
107 |
1 2 103 104 93 105 106
|
dssmapfv3d |
|- ( ( ( ph /\ B e. _V ) /\ s e. ~P B ) -> ( ( D ` I ) ` s ) = ( B \ ( I ` ( B \ s ) ) ) ) |
108 |
102 107
|
sylan2 |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( D ` I ) ` s ) = ( B \ ( I ` ( B \ s ) ) ) ) |
109 |
|
simpr |
|- ( ( s e. ~P B /\ t e. ~P B ) -> t e. ~P B ) |
110 |
|
simplr |
|- ( ( ( ph /\ B e. _V ) /\ t e. ~P B ) -> B e. _V ) |
111 |
71
|
ad2antrr |
|- ( ( ( ph /\ B e. _V ) /\ t e. ~P B ) -> I e. ( ~P B ^m ~P B ) ) |
112 |
|
simpr |
|- ( ( ( ph /\ B e. _V ) /\ t e. ~P B ) -> t e. ~P B ) |
113 |
|
eqid |
|- ( ( D ` I ) ` t ) = ( ( D ` I ) ` t ) |
114 |
1 2 110 111 93 112 113
|
dssmapfv3d |
|- ( ( ( ph /\ B e. _V ) /\ t e. ~P B ) -> ( ( D ` I ) ` t ) = ( B \ ( I ` ( B \ t ) ) ) ) |
115 |
109 114
|
sylan2 |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( D ` I ) ` t ) = ( B \ ( I ` ( B \ t ) ) ) ) |
116 |
108 115
|
uneq12d |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) = ( ( B \ ( I ` ( B \ s ) ) ) u. ( B \ ( I ` ( B \ t ) ) ) ) ) |
117 |
|
difindi |
|- ( B \ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) = ( ( B \ ( I ` ( B \ s ) ) ) u. ( B \ ( I ` ( B \ t ) ) ) ) |
118 |
116 117
|
eqtr4di |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) = ( B \ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) |
119 |
101 118
|
eqeq12d |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( ( D ` I ) ` ( s u. t ) ) = ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) <-> ( B \ ( I ` ( B \ ( s u. t ) ) ) ) = ( B \ ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) ) ) ) |
120 |
|
simpll |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ph ) |
121 |
1 2 3
|
ntrclsfv1 |
|- ( ph -> ( D ` I ) = K ) |
122 |
|
fveq1 |
|- ( ( D ` I ) = K -> ( ( D ` I ) ` ( s u. t ) ) = ( K ` ( s u. t ) ) ) |
123 |
|
fveq1 |
|- ( ( D ` I ) = K -> ( ( D ` I ) ` s ) = ( K ` s ) ) |
124 |
|
fveq1 |
|- ( ( D ` I ) = K -> ( ( D ` I ) ` t ) = ( K ` t ) ) |
125 |
123 124
|
uneq12d |
|- ( ( D ` I ) = K -> ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) |
126 |
122 125
|
eqeq12d |
|- ( ( D ` I ) = K -> ( ( ( D ` I ) ` ( s u. t ) ) = ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) <-> ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
127 |
120 121 126
|
3syl |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( ( D ` I ) ` ( s u. t ) ) = ( ( ( D ` I ) ` s ) u. ( ( D ` I ) ` t ) ) <-> ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
128 |
90 119 127
|
3bitr2d |
|- ( ( ( ph /\ B e. _V ) /\ ( s e. ~P B /\ t e. ~P B ) ) -> ( ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) <-> ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
129 |
68 69 70 128
|
syl12anc |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` ( B \ ( s u. t ) ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` ( B \ t ) ) ) <-> ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
130 |
66 129
|
bitrd |
|- ( ( ( ph /\ s e. ~P B ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) <-> ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
131 |
42 58 130
|
ralxfrd2 |
|- ( ( ph /\ s e. ~P B ) -> ( A. b e. ~P B ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) <-> A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
132 |
131
|
3adant3 |
|- ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) -> ( A. b e. ~P B ( I ` ( ( B \ s ) i^i b ) ) = ( ( I ` ( B \ s ) ) i^i ( I ` b ) ) <-> A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
133 |
39 132
|
bitrd |
|- ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) -> ( A. b e. ~P B ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) <-> A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
134 |
18 32 133
|
ralxfrd2 |
|- ( ph -> ( A. a e. ~P B A. b e. ~P B ( I ` ( a i^i b ) ) = ( ( I ` a ) i^i ( I ` b ) ) <-> A. s e. ~P B A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |
135 |
14 134
|
syl5bb |
|- ( ph -> ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( K ` ( s u. t ) ) = ( ( K ` s ) u. ( K ` t ) ) ) ) |