Metamath Proof Explorer


Theorem ptuncnv

Description: Exhibit the converse function of the map G which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015) (Proof shortened by Mario Carneiro, 23-Aug-2015)

Ref Expression
Hypotheses ptunhmeo.x
|- X = U. K
ptunhmeo.y
|- Y = U. L
ptunhmeo.j
|- J = ( Xt_ ` F )
ptunhmeo.k
|- K = ( Xt_ ` ( F |` A ) )
ptunhmeo.l
|- L = ( Xt_ ` ( F |` B ) )
ptunhmeo.g
|- G = ( x e. X , y e. Y |-> ( x u. y ) )
ptunhmeo.c
|- ( ph -> C e. V )
ptunhmeo.f
|- ( ph -> F : C --> Top )
ptunhmeo.u
|- ( ph -> C = ( A u. B ) )
ptunhmeo.i
|- ( ph -> ( A i^i B ) = (/) )
Assertion ptuncnv
|- ( ph -> `' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) )

Proof

Step Hyp Ref Expression
1 ptunhmeo.x
 |-  X = U. K
2 ptunhmeo.y
 |-  Y = U. L
3 ptunhmeo.j
 |-  J = ( Xt_ ` F )
4 ptunhmeo.k
 |-  K = ( Xt_ ` ( F |` A ) )
5 ptunhmeo.l
 |-  L = ( Xt_ ` ( F |` B ) )
6 ptunhmeo.g
 |-  G = ( x e. X , y e. Y |-> ( x u. y ) )
7 ptunhmeo.c
 |-  ( ph -> C e. V )
8 ptunhmeo.f
 |-  ( ph -> F : C --> Top )
9 ptunhmeo.u
 |-  ( ph -> C = ( A u. B ) )
10 ptunhmeo.i
 |-  ( ph -> ( A i^i B ) = (/) )
11 vex
 |-  x e. _V
12 vex
 |-  y e. _V
13 11 12 op1std
 |-  ( w = <. x , y >. -> ( 1st ` w ) = x )
14 11 12 op2ndd
 |-  ( w = <. x , y >. -> ( 2nd ` w ) = y )
15 13 14 uneq12d
 |-  ( w = <. x , y >. -> ( ( 1st ` w ) u. ( 2nd ` w ) ) = ( x u. y ) )
16 15 mpompt
 |-  ( w e. ( X X. Y ) |-> ( ( 1st ` w ) u. ( 2nd ` w ) ) ) = ( x e. X , y e. Y |-> ( x u. y ) )
17 6 16 eqtr4i
 |-  G = ( w e. ( X X. Y ) |-> ( ( 1st ` w ) u. ( 2nd ` w ) ) )
18 xp1st
 |-  ( w e. ( X X. Y ) -> ( 1st ` w ) e. X )
19 18 adantl
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 1st ` w ) e. X )
20 ixpeq2
 |-  ( A. k e. A U. ( ( F |` A ) ` k ) = U. ( F ` k ) -> X_ k e. A U. ( ( F |` A ) ` k ) = X_ k e. A U. ( F ` k ) )
21 fvres
 |-  ( k e. A -> ( ( F |` A ) ` k ) = ( F ` k ) )
22 21 unieqd
 |-  ( k e. A -> U. ( ( F |` A ) ` k ) = U. ( F ` k ) )
23 20 22 mprg
 |-  X_ k e. A U. ( ( F |` A ) ` k ) = X_ k e. A U. ( F ` k )
24 ssun1
 |-  A C_ ( A u. B )
25 24 9 sseqtrrid
 |-  ( ph -> A C_ C )
26 7 25 ssexd
 |-  ( ph -> A e. _V )
27 8 25 fssresd
 |-  ( ph -> ( F |` A ) : A --> Top )
28 4 ptuni
 |-  ( ( A e. _V /\ ( F |` A ) : A --> Top ) -> X_ k e. A U. ( ( F |` A ) ` k ) = U. K )
29 26 27 28 syl2anc
 |-  ( ph -> X_ k e. A U. ( ( F |` A ) ` k ) = U. K )
30 23 29 syl5eqr
 |-  ( ph -> X_ k e. A U. ( F ` k ) = U. K )
31 30 1 eqtr4di
 |-  ( ph -> X_ k e. A U. ( F ` k ) = X )
32 31 adantr
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> X_ k e. A U. ( F ` k ) = X )
33 19 32 eleqtrrd
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 1st ` w ) e. X_ k e. A U. ( F ` k ) )
34 xp2nd
 |-  ( w e. ( X X. Y ) -> ( 2nd ` w ) e. Y )
35 34 adantl
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 2nd ` w ) e. Y )
36 9 eqcomd
 |-  ( ph -> ( A u. B ) = C )
37 uneqdifeq
 |-  ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C <-> ( C \ A ) = B ) )
38 25 10 37 syl2anc
 |-  ( ph -> ( ( A u. B ) = C <-> ( C \ A ) = B ) )
39 36 38 mpbid
 |-  ( ph -> ( C \ A ) = B )
40 39 ixpeq1d
 |-  ( ph -> X_ k e. ( C \ A ) U. ( F ` k ) = X_ k e. B U. ( F ` k ) )
41 ixpeq2
 |-  ( A. k e. B U. ( ( F |` B ) ` k ) = U. ( F ` k ) -> X_ k e. B U. ( ( F |` B ) ` k ) = X_ k e. B U. ( F ` k ) )
42 fvres
 |-  ( k e. B -> ( ( F |` B ) ` k ) = ( F ` k ) )
43 42 unieqd
 |-  ( k e. B -> U. ( ( F |` B ) ` k ) = U. ( F ` k ) )
44 41 43 mprg
 |-  X_ k e. B U. ( ( F |` B ) ` k ) = X_ k e. B U. ( F ` k )
45 ssun2
 |-  B C_ ( A u. B )
46 45 9 sseqtrrid
 |-  ( ph -> B C_ C )
47 7 46 ssexd
 |-  ( ph -> B e. _V )
48 8 46 fssresd
 |-  ( ph -> ( F |` B ) : B --> Top )
49 5 ptuni
 |-  ( ( B e. _V /\ ( F |` B ) : B --> Top ) -> X_ k e. B U. ( ( F |` B ) ` k ) = U. L )
50 47 48 49 syl2anc
 |-  ( ph -> X_ k e. B U. ( ( F |` B ) ` k ) = U. L )
51 44 50 syl5eqr
 |-  ( ph -> X_ k e. B U. ( F ` k ) = U. L )
52 51 2 eqtr4di
 |-  ( ph -> X_ k e. B U. ( F ` k ) = Y )
53 40 52 eqtrd
 |-  ( ph -> X_ k e. ( C \ A ) U. ( F ` k ) = Y )
54 53 adantr
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> X_ k e. ( C \ A ) U. ( F ` k ) = Y )
55 35 54 eleqtrrd
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 2nd ` w ) e. X_ k e. ( C \ A ) U. ( F ` k ) )
56 25 adantr
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> A C_ C )
57 undifixp
 |-  ( ( ( 1st ` w ) e. X_ k e. A U. ( F ` k ) /\ ( 2nd ` w ) e. X_ k e. ( C \ A ) U. ( F ` k ) /\ A C_ C ) -> ( ( 1st ` w ) u. ( 2nd ` w ) ) e. X_ k e. C U. ( F ` k ) )
58 33 55 56 57 syl3anc
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( ( 1st ` w ) u. ( 2nd ` w ) ) e. X_ k e. C U. ( F ` k ) )
59 3 ptuni
 |-  ( ( C e. V /\ F : C --> Top ) -> X_ k e. C U. ( F ` k ) = U. J )
60 7 8 59 syl2anc
 |-  ( ph -> X_ k e. C U. ( F ` k ) = U. J )
61 60 adantr
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> X_ k e. C U. ( F ` k ) = U. J )
62 58 61 eleqtrd
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( ( 1st ` w ) u. ( 2nd ` w ) ) e. U. J )
63 25 adantr
 |-  ( ( ph /\ z e. U. J ) -> A C_ C )
64 60 eleq2d
 |-  ( ph -> ( z e. X_ k e. C U. ( F ` k ) <-> z e. U. J ) )
65 64 biimpar
 |-  ( ( ph /\ z e. U. J ) -> z e. X_ k e. C U. ( F ` k ) )
66 resixp
 |-  ( ( A C_ C /\ z e. X_ k e. C U. ( F ` k ) ) -> ( z |` A ) e. X_ k e. A U. ( F ` k ) )
67 63 65 66 syl2anc
 |-  ( ( ph /\ z e. U. J ) -> ( z |` A ) e. X_ k e. A U. ( F ` k ) )
68 31 adantr
 |-  ( ( ph /\ z e. U. J ) -> X_ k e. A U. ( F ` k ) = X )
69 67 68 eleqtrd
 |-  ( ( ph /\ z e. U. J ) -> ( z |` A ) e. X )
70 46 adantr
 |-  ( ( ph /\ z e. U. J ) -> B C_ C )
71 resixp
 |-  ( ( B C_ C /\ z e. X_ k e. C U. ( F ` k ) ) -> ( z |` B ) e. X_ k e. B U. ( F ` k ) )
72 70 65 71 syl2anc
 |-  ( ( ph /\ z e. U. J ) -> ( z |` B ) e. X_ k e. B U. ( F ` k ) )
73 52 adantr
 |-  ( ( ph /\ z e. U. J ) -> X_ k e. B U. ( F ` k ) = Y )
74 72 73 eleqtrd
 |-  ( ( ph /\ z e. U. J ) -> ( z |` B ) e. Y )
75 69 74 opelxpd
 |-  ( ( ph /\ z e. U. J ) -> <. ( z |` A ) , ( z |` B ) >. e. ( X X. Y ) )
76 eqop
 |-  ( w e. ( X X. Y ) -> ( w = <. ( z |` A ) , ( z |` B ) >. <-> ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) ) )
77 76 ad2antrl
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( w = <. ( z |` A ) , ( z |` B ) >. <-> ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) ) )
78 65 adantrl
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> z e. X_ k e. C U. ( F ` k ) )
79 ixpfn
 |-  ( z e. X_ k e. C U. ( F ` k ) -> z Fn C )
80 fnresdm
 |-  ( z Fn C -> ( z |` C ) = z )
81 78 79 80 3syl
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( z |` C ) = z )
82 9 reseq2d
 |-  ( ph -> ( z |` C ) = ( z |` ( A u. B ) ) )
83 82 adantr
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( z |` C ) = ( z |` ( A u. B ) ) )
84 81 83 eqtr3d
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> z = ( z |` ( A u. B ) ) )
85 resundi
 |-  ( z |` ( A u. B ) ) = ( ( z |` A ) u. ( z |` B ) )
86 84 85 eqtrdi
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> z = ( ( z |` A ) u. ( z |` B ) ) )
87 uneq12
 |-  ( ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) -> ( ( 1st ` w ) u. ( 2nd ` w ) ) = ( ( z |` A ) u. ( z |` B ) ) )
88 87 eqeq2d
 |-  ( ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) -> ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) <-> z = ( ( z |` A ) u. ( z |` B ) ) ) )
89 86 88 syl5ibrcom
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) -> z = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) )
90 ixpfn
 |-  ( ( 1st ` w ) e. X_ k e. A U. ( F ` k ) -> ( 1st ` w ) Fn A )
91 33 90 syl
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 1st ` w ) Fn A )
92 91 adantrr
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 1st ` w ) Fn A )
93 dffn2
 |-  ( ( 1st ` w ) Fn A <-> ( 1st ` w ) : A --> _V )
94 92 93 sylib
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 1st ` w ) : A --> _V )
95 52 adantr
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> X_ k e. B U. ( F ` k ) = Y )
96 35 95 eleqtrrd
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 2nd ` w ) e. X_ k e. B U. ( F ` k ) )
97 ixpfn
 |-  ( ( 2nd ` w ) e. X_ k e. B U. ( F ` k ) -> ( 2nd ` w ) Fn B )
98 96 97 syl
 |-  ( ( ph /\ w e. ( X X. Y ) ) -> ( 2nd ` w ) Fn B )
99 98 adantrr
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 2nd ` w ) Fn B )
100 dffn2
 |-  ( ( 2nd ` w ) Fn B <-> ( 2nd ` w ) : B --> _V )
101 99 100 sylib
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 2nd ` w ) : B --> _V )
102 res0
 |-  ( ( 1st ` w ) |` (/) ) = (/)
103 res0
 |-  ( ( 2nd ` w ) |` (/) ) = (/)
104 102 103 eqtr4i
 |-  ( ( 1st ` w ) |` (/) ) = ( ( 2nd ` w ) |` (/) )
105 10 adantr
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( A i^i B ) = (/) )
106 105 reseq2d
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( 1st ` w ) |` ( A i^i B ) ) = ( ( 1st ` w ) |` (/) ) )
107 105 reseq2d
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( 2nd ` w ) |` ( A i^i B ) ) = ( ( 2nd ` w ) |` (/) ) )
108 104 106 107 3eqtr4a
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( 1st ` w ) |` ( A i^i B ) ) = ( ( 2nd ` w ) |` ( A i^i B ) ) )
109 fresaunres1
 |-  ( ( ( 1st ` w ) : A --> _V /\ ( 2nd ` w ) : B --> _V /\ ( ( 1st ` w ) |` ( A i^i B ) ) = ( ( 2nd ` w ) |` ( A i^i B ) ) ) -> ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) = ( 1st ` w ) )
110 94 101 108 109 syl3anc
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) = ( 1st ` w ) )
111 110 eqcomd
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 1st ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) )
112 fresaunres2
 |-  ( ( ( 1st ` w ) : A --> _V /\ ( 2nd ` w ) : B --> _V /\ ( ( 1st ` w ) |` ( A i^i B ) ) = ( ( 2nd ` w ) |` ( A i^i B ) ) ) -> ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) = ( 2nd ` w ) )
113 94 101 108 112 syl3anc
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) = ( 2nd ` w ) )
114 113 eqcomd
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( 2nd ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) )
115 111 114 jca
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( 1st ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) /\ ( 2nd ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) ) )
116 reseq1
 |-  ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( z |` A ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) )
117 116 eqeq2d
 |-  ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( ( 1st ` w ) = ( z |` A ) <-> ( 1st ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) ) )
118 reseq1
 |-  ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( z |` B ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) )
119 118 eqeq2d
 |-  ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( ( 2nd ` w ) = ( z |` B ) <-> ( 2nd ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) ) )
120 117 119 anbi12d
 |-  ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) <-> ( ( 1st ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` A ) /\ ( 2nd ` w ) = ( ( ( 1st ` w ) u. ( 2nd ` w ) ) |` B ) ) ) )
121 115 120 syl5ibrcom
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( z = ( ( 1st ` w ) u. ( 2nd ` w ) ) -> ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) ) )
122 89 121 impbid
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( ( ( 1st ` w ) = ( z |` A ) /\ ( 2nd ` w ) = ( z |` B ) ) <-> z = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) )
123 77 122 bitrd
 |-  ( ( ph /\ ( w e. ( X X. Y ) /\ z e. U. J ) ) -> ( w = <. ( z |` A ) , ( z |` B ) >. <-> z = ( ( 1st ` w ) u. ( 2nd ` w ) ) ) )
124 17 62 75 123 f1ocnv2d
 |-  ( ph -> ( G : ( X X. Y ) -1-1-onto-> U. J /\ `' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) ) )
125 124 simprd
 |-  ( ph -> `' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) )