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Theorem tgqioo 21175
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 5335 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  C_  ran  (,)
3 ioof 11628 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
4 ffn 5718 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
53, 4ax-mp 5 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
6 simpll 753 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  e.  RR* )
7 elioo1 11575 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
z  e.  ( x (,) y )  <->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) ) )
87biimpa 484 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) )
98simp1d 1007 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  e.  RR* )
108simp2d 1008 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  <  z )
11 qbtwnxr 11405 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  z  e.  RR*  /\  x  < 
z )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
126, 9, 10, 11syl3anc 1227 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
13 simplr 754 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  y  e.  RR* )
148simp3d 1009 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  <  y )
15 qbtwnxr 11405 . . . . . . . . . 10  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  z  < 
y )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
169, 13, 14, 15syl3anc 1227 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
17 reeanv 3009 . . . . . . . . . 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 6281 . . . . . . . . . . . . . 14  |-  ( u (,) v )  =  ( (,) `  <. u ,  v >. )
19 opelxpi 5018 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  QQ  /\  v  e.  QQ )  -> 
<. u ,  v >.  e.  ( QQ  X.  QQ ) )
20193ad2ant2 1017 . . . . . . . . . . . . . . 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 5720 . . . . . . . . . . . . . . . . 17  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
223, 21ax-mp 5 . . . . . . . . . . . . . . . 16  |-  Fun  (,)
23 qssre 11198 . . . . . . . . . . . . . . . . . . 19  |-  QQ  C_  RR
24 ressxr 9637 . . . . . . . . . . . . . . . . . . 19  |-  RR  C_  RR*
2523, 24sstri 3496 . . . . . . . . . . . . . . . . . 18  |-  QQ  C_  RR*
26 xpss12 5095 . . . . . . . . . . . . . . . . . 18  |-  ( ( QQ  C_  RR*  /\  QQ  C_ 
RR* )  ->  ( QQ  X.  QQ )  C_  ( RR*  X.  RR* )
)
2725, 25, 26mp2an 672 . . . . . . . . . . . . . . . . 17  |-  ( QQ 
X.  QQ )  C_  ( RR*  X.  RR* )
283fdmi 5723 . . . . . . . . . . . . . . . . 17  |-  dom  (,)  =  ( RR*  X.  RR* )
2927, 28sseqtr4i 3520 . . . . . . . . . . . . . . . 16  |-  ( QQ 
X.  QQ )  C_  dom  (,)
30 funfvima2 6130 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  (,)  /\  ( QQ  X.  QQ )  C_  dom  (,) )  ->  ( <. u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) ) )
3122, 29, 30mp2an 672 . . . . . . . . . . . . . . 15  |-  ( <.
u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) )
3220, 31syl 16 . . . . . . . . . . . . . 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 2533 . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . . 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 3485 . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . . 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 3485 . . . . . . . . . . . . . . 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 11575 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  RR*  /\  v  e.  RR* )  ->  (
z  e.  ( u (,) v )  <->  ( z  e.  RR*  /\  u  < 
z  /\  z  <  v ) ) )
4238, 40, 41syl2anc 661 . . . . . . . . . . . . . 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 1178 . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . . . . 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 11361 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  u  e.  RR* )  ->  (
x  <  u  ->  x  <_  u ) )
4744, 38, 46syl2anc 661 . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . 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 11570 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  x  <_  u )  ->  (
u (,) v ) 
C_  ( x (,) v ) )
5044, 48, 49syl2anc 661 . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . . . 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 11361 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  RR*  /\  y  e.  RR* )  ->  (
v  <  y  ->  v  <_  y ) )
5440, 51, 53syl2anc 661 . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . 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 11571 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  v  <_  y )  ->  (
x (,) v ) 
C_  ( x (,) y ) )
5751, 55, 56syl2anc 661 . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . 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 2514 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
z  e.  w  <->  z  e.  ( u (,) v
) ) )
60 sseq1 3508 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
w  C_  ( x (,) y )  <->  ( u (,) v )  C_  (
x (,) y ) ) )
6159, 60anbi12d 710 . . . . . . . . . . . . . 14  |-  ( w  =  ( u (,) v )  ->  (
( z  e.  w  /\  w  C_  ( x (,) y ) )  <-> 
( z  e.  ( u (,) v )  /\  ( u (,) v )  C_  (
x (,) y ) ) ) )
6261rspcev 3194 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . 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 1194 . . . . . . . . . . 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 2939 . . . . . . . . . 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 218 . . . . . . . . 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 679 . . . . . . . 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 2855 . . . . . . 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 21136 . . . . . . . 8  |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
70 eltg2b 19330 . . . . . . . 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 5 . . . . . . 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 212 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7372rgen2a 2868 . . . . 5  |-  A. x  e.  RR*  A. y  e. 
RR*  ( x (,) y )  e.  (
topGen `  ( (,) " ( QQ  X.  QQ ) ) )
74 ffnov 6388 . . . . 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 918 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
76 frn 5724 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  ->  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7775, 76ax-mp 5 . . 3  |-  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
78 2basgen 19362 . . 3  |-  ( ( ( (,) " ( QQ  X.  QQ ) ) 
C_  ran  (,)  /\  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )  ->  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen `  ran  (,) )
)
792, 77, 78mp2an 672 . 2  |-  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen ` 
ran  (,) )
801, 79eqtr2i 2471 1  |-  ( topGen ` 
ran  (,) )  =  Q
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792    C_ wss 3459   ~Pcpw 3994   <.cop 4017   class class class wbr 4434    X. cxp 4984   dom cdm 4986   ran crn 4987   "cima 4989   Fun wfun 5569    Fn wfn 5570   -->wf 5571   ` cfv 5575  (class class class)co 6278   RRcr 9491   RR*cxr 9627    < clt 9628    <_ cle 9629   QQcq 11188   (,)cioo 11535   topGenctg 14709   TopBasesctb 19268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-sup 7900  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-n0 10799  df-z 10868  df-uz 11088  df-q 11189  df-ioo 11539  df-topgen 14715  df-bases 19271
This theorem is referenced by:  re2ndc  21176  opnmblALT  21882  mbfimaopnlem  21932
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