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Theorem tgqioo 18306
Description: The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
Hypothesis
Ref Expression
tgqioo.1  |-  Q  =  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
Assertion
Ref Expression
tgqioo  |-  ( topGen ` 
ran  (,) )  =  Q

Proof of Theorem tgqioo
Dummy variables  v  u  w  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgqioo.1 . 2  |-  Q  =  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
2 imassrn 5025 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  C_  ran  (,)
3 ioof 10741 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
4 ffn 5389 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
53, 4ax-mp 8 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
6 simpll 730 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  e.  RR* )
7 elioo1 10696 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
z  e.  ( x (,) y )  <->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) ) )
87biimpa 470 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) )
98simp1d 967 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  e.  RR* )
108simp2d 968 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  <  z )
11 qbtwnxr 10527 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  z  e.  RR*  /\  x  < 
z )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
126, 9, 10, 11syl3anc 1182 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
13 simplr 731 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  y  e.  RR* )
148simp3d 969 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  <  y )
15 qbtwnxr 10527 . . . . . . . . . 10  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  z  < 
y )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
169, 13, 14, 15syl3anc 1182 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
17 reeanv 2707 . . . . . . . . . 10  |-  ( E. u  e.  QQ  E. v  e.  QQ  (
( x  <  u  /\  u  <  z )  /\  ( z  < 
v  /\  v  <  y ) )  <->  ( E. u  e.  QQ  (
x  <  u  /\  u  <  z )  /\  E. v  e.  QQ  (
z  <  v  /\  v  <  y ) ) )
18 df-ov 5861 . . . . . . . . . . . . . 14  |-  ( u (,) v )  =  ( (,) `  <. u ,  v >. )
19 opelxpi 4721 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  QQ  /\  v  e.  QQ )  -> 
<. u ,  v >.  e.  ( QQ  X.  QQ ) )
20193ad2ant2 977 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  <. u ,  v
>.  e.  ( QQ  X.  QQ ) )
21 ffun 5391 . . . . . . . . . . . . . . . . 17  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
223, 21ax-mp 8 . . . . . . . . . . . . . . . 16  |-  Fun  (,)
23 qssre 10326 . . . . . . . . . . . . . . . . . . 19  |-  QQ  C_  RR
24 ressxr 8876 . . . . . . . . . . . . . . . . . . 19  |-  RR  C_  RR*
2523, 24sstri 3188 . . . . . . . . . . . . . . . . . 18  |-  QQ  C_  RR*
26 xpss12 4792 . . . . . . . . . . . . . . . . . 18  |-  ( ( QQ  C_  RR*  /\  QQ  C_ 
RR* )  ->  ( QQ  X.  QQ )  C_  ( RR*  X.  RR* )
)
2725, 25, 26mp2an 653 . . . . . . . . . . . . . . . . 17  |-  ( QQ 
X.  QQ )  C_  ( RR*  X.  RR* )
283fdmi 5394 . . . . . . . . . . . . . . . . 17  |-  dom  (,)  =  ( RR*  X.  RR* )
2927, 28sseqtr4i 3211 . . . . . . . . . . . . . . . 16  |-  ( QQ 
X.  QQ )  C_  dom  (,)
30 funfvima2 5754 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  (,)  /\  ( QQ  X.  QQ )  C_  dom  (,) )  ->  ( <. u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) ) )
3122, 29, 30mp2an 653 . . . . . . . . . . . . . . 15  |-  ( <.
u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) )
3220, 31syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) )
3318, 32syl5eqel 2367 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  e.  ( (,) " ( QQ 
X.  QQ ) ) )
3493ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  e.  RR* )
35 simp3lr 1027 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  <  z
)
36 simp3rl 1028 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  <  v
)
37 simp2l 981 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  e.  QQ )
3825, 37sseldi 3178 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  e.  RR* )
39 simp2r 982 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  e.  QQ )
4025, 39sseldi 3178 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  e.  RR* )
41 elioo1 10696 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  RR*  /\  v  e.  RR* )  ->  (
z  e.  ( u (,) v )  <->  ( z  e.  RR*  /\  u  < 
z  /\  z  <  v ) ) )
4238, 40, 41syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( z  e.  ( u (,) v
)  <->  ( z  e. 
RR*  /\  u  <  z  /\  z  <  v
) ) )
4334, 35, 36, 42mpbir3and 1135 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  e.  ( u (,) v ) )
4463ad2ant1 976 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  e.  RR* )
45 simp3ll 1026 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  <  u
)
46 xrltle 10483 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  u  e.  RR* )  ->  (
x  <  u  ->  x  <_  u ) )
4744, 38, 46syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( x  < 
u  ->  x  <_  u ) )
4845, 47mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  <_  u
)
49 iooss1 10691 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  x  <_  u )  ->  (
u (,) v ) 
C_  ( x (,) v ) )
5044, 48, 49syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  C_  (
x (,) v ) )
51133ad2ant1 976 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  y  e.  RR* )
52 simp3rr 1029 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  <  y
)
53 xrltle 10483 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  RR*  /\  y  e.  RR* )  ->  (
v  <  y  ->  v  <_  y ) )
5440, 51, 53syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( v  < 
y  ->  v  <_  y ) )
5552, 54mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  <_  y
)
56 iooss2 10692 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  v  <_  y )  ->  (
x (,) v ) 
C_  ( x (,) y ) )
5751, 55, 56syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( x (,) v )  C_  (
x (,) y ) )
5850, 57sstrd 3189 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  C_  (
x (,) y ) )
59 eleq2 2344 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
z  e.  w  <->  z  e.  ( u (,) v
) ) )
60 sseq1 3199 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
w  C_  ( x (,) y )  <->  ( u (,) v )  C_  (
x (,) y ) ) )
6159, 60anbi12d 691 . . . . . . . . . . . . . 14  |-  ( w  =  ( u (,) v )  ->  (
( z  e.  w  /\  w  C_  ( x (,) y ) )  <-> 
( z  e.  ( u (,) v )  /\  ( u (,) v )  C_  (
x (,) y ) ) ) )
6261rspcev 2884 . . . . . . . . . . . . 13  |-  ( ( ( u (,) v
)  e.  ( (,) " ( QQ  X.  QQ ) )  /\  (
z  e.  ( u (,) v )  /\  ( u (,) v
)  C_  ( x (,) y ) ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
6333, 43, 58, 62syl12anc 1180 . . . . . . . . . . . 12  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
64633exp 1150 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( (
u  e.  QQ  /\  v  e.  QQ )  ->  ( ( ( x  <  u  /\  u  <  z )  /\  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) ) )
6564rexlimdvv 2673 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( E. u  e.  QQ  E. v  e.  QQ  ( ( x  <  u  /\  u  <  z )  /\  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
6617, 65syl5bir 209 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( ( E. u  e.  QQ  ( x  <  u  /\  u  <  z )  /\  E. v  e.  QQ  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
6712, 16, 66mp2and 660 . . . . . . . 8  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
6867ralrimiva 2626 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
69 qtopbas 18268 . . . . . . . 8  |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
70 eltg2b 16697 . . . . . . . 8  |-  ( ( (,) " ( QQ 
X.  QQ ) )  e.  TopBases  ->  ( ( x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
7169, 70ax-mp 8 . . . . . . 7  |-  ( ( x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
7268, 71sylibr 203 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7372rgen2a 2609 . . . . 5  |-  A. x  e.  RR*  A. y  e. 
RR*  ( x (,) y )  e.  (
topGen `  ( (,) " ( QQ  X.  QQ ) ) )
74 ffnov 5948 . . . . 5  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  ( (,)  Fn  ( RR*  X.  RR* )  /\  A. x  e.  RR*  A. y  e.  RR*  (
x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) ) )
755, 73, 74mpbir2an 886 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
76 frn 5395 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  ->  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7775, 76ax-mp 8 . . 3  |-  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
78 2basgen 16728 . . 3  |-  ( ( ( (,) " ( QQ  X.  QQ ) ) 
C_  ran  (,)  /\  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )  ->  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen `  ran  (,) )
)
792, 77, 78mp2an 653 . 2  |-  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen ` 
ran  (,) )
801, 79eqtr2i 2304 1  |-  ( topGen ` 
ran  (,) )  =  Q
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868   QQcq 10316   (,)cioo 10656   topGenctg 13342   TopBasesctb 16635
This theorem is referenced by:  re2ndc  18307  opnmblALT  18958  mbfimaopnlem  19010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660  df-topgen 13344  df-bases 16638
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