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Theorem 1stccn 20409
Description: A mapping  X --> Y, where  X is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence  f (
n ) converging to  x, its image under  F converges to  F ( x ). (Contributed by Mario Carneiro, 7-Sep-2015.)
Hypotheses
Ref Expression
1stccnp.1  |-  ( ph  ->  J  e.  1stc )
1stccnp.2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
1stccnp.3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
1stccn.7  |-  ( ph  ->  F : X --> Y )
Assertion
Ref Expression
1stccn  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <->  A. f ( f : NN --> X  ->  A. x
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
Distinct variable groups:    x, f, F    f, J, x    ph, f, x    f, K, x    f, X, x    f, Y, x

Proof of Theorem 1stccn
StepHypRef Expression
1 1stccnp.2 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 1stccnp.3 . . . 4  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
3 cncnp 20227 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) ) )
41, 2, 3syl2anc 665 . . 3  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. x  e.  X  F  e.  ( ( J  CnP  K
) `  x )
) ) )
5 1stccn.7 . . . 4  |-  ( ph  ->  F : X --> Y )
65biantrurd 510 . . 3  |-  ( ph  ->  ( A. x  e.  X  F  e.  ( ( J  CnP  K
) `  x )  <->  ( F : X --> Y  /\  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) ) )
74, 6bitr4d 259 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <->  A. x  e.  X  F  e.  ( ( J  CnP  K ) `  x ) ) )
8 1stccnp.1 . . . . . 6  |-  ( ph  ->  J  e.  1stc )
98adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  1stc )
101adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  J  e.  (TopOn `  X )
)
112adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  K  e.  (TopOn `  Y )
)
12 simpr 462 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
139, 10, 11, 121stccnp 20408 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  x )  <->  ( F : X --> Y  /\  A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
145adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> Y )
1514biantrurd 510 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( F : X --> Y  /\  A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
1613, 15bitr4d 259 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  x )  <->  A. f
( ( f : NN --> X  /\  f
( ~~> t `  J
) x )  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) )
1716ralbidva 2868 . 2  |-  ( ph  ->  ( A. x  e.  X  F  e.  ( ( J  CnP  K
) `  x )  <->  A. x  e.  X  A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) )
18 ralcom4 3106 . . 3  |-  ( A. x  e.  X  A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <->  A. f A. x  e.  X  ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) )
19 impexp 447 . . . . . . 7  |-  ( ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( f : NN --> X  ->  ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) )
2019ralbii 2863 . . . . . 6  |-  ( A. x  e.  X  (
( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <->  A. x  e.  X  ( f : NN --> X  ->  ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) )
21 r19.21v 2837 . . . . . 6  |-  ( A. x  e.  X  (
f : NN --> X  -> 
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) )  <->  ( f : NN --> X  ->  A. x  e.  X  ( f
( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) )
2220, 21bitri 252 . . . . 5  |-  ( A. x  e.  X  (
( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( f : NN --> X  ->  A. x  e.  X  ( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) )
23 df-ral 2787 . . . . . . 7  |-  ( A. x  e.  X  (
f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) )  <->  A. x ( x  e.  X  ->  ( f
( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) )
24 lmcl 20244 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  f
( ~~> t `  J
) x )  ->  x  e.  X )
251, 24sylan 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  f ( ~~> t `  J )
x )  ->  x  e.  X )
2625ex 435 . . . . . . . . . . 11  |-  ( ph  ->  ( f ( ~~> t `  J ) x  ->  x  e.  X )
)
2726pm4.71rd 639 . . . . . . . . . 10  |-  ( ph  ->  ( f ( ~~> t `  J ) x  <->  ( x  e.  X  /\  f
( ~~> t `  J
) x ) ) )
2827imbi1d 318 . . . . . . . . 9  |-  ( ph  ->  ( ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( ( x  e.  X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) )
29 impexp 447 . . . . . . . . 9  |-  ( ( ( x  e.  X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( x  e.  X  ->  ( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) )
3028, 29syl6rbb 265 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  X  ->  ( f
( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) )  <-> 
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) )
3130albidv 1760 . . . . . . 7  |-  ( ph  ->  ( A. x ( x  e.  X  -> 
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) )  <->  A. x ( f ( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) )
3223, 31syl5bb 260 . . . . . 6  |-  ( ph  ->  ( A. x  e.  X  ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <->  A. x ( f ( ~~> t `  J ) x  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) ) ) )
3332imbi2d 317 . . . . 5  |-  ( ph  ->  ( ( f : NN --> X  ->  A. x  e.  X  ( f
( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) )  <-> 
( f : NN --> X  ->  A. x ( f ( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) ) )
3422, 33syl5bb 260 . . . 4  |-  ( ph  ->  ( A. x  e.  X  ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <-> 
( f : NN --> X  ->  A. x ( f ( ~~> t `  J
) x  ->  ( F  o.  f )
( ~~> t `  K
) ( F `  x ) ) ) ) )
3534albidv 1760 . . 3  |-  ( ph  ->  ( A. f A. x  e.  X  (
( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <->  A. f ( f : NN --> X  ->  A. x
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
3618, 35syl5bb 260 . 2  |-  ( ph  ->  ( A. x  e.  X  A. f ( ( f : NN --> X  /\  f ( ~~> t `  J ) x )  ->  ( F  o.  f ) ( ~~> t `  K ) ( F `
 x ) )  <->  A. f ( f : NN --> X  ->  A. x
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
377, 17, 363bitrd 282 1  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <->  A. f ( f : NN --> X  ->  A. x
( f ( ~~> t `  J ) x  -> 
( F  o.  f
) ( ~~> t `  K ) ( F `
 x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    e. wcel 1870   A.wral 2782   class class class wbr 4426    o. ccom 4858   -->wf 5597   ` cfv 5601  (class class class)co 6305   NNcn 10609  TopOnctopon 19849    Cn ccn 20171    CnP ccnp 20172   ~~> tclm 20173   1stcc1stc 20383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cc 8863  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-topgen 15301  df-top 19852  df-topon 19854  df-cld 19965  df-ntr 19966  df-cls 19967  df-cn 20174  df-cnp 20175  df-lm 20176  df-1stc 20385
This theorem is referenced by:  metcn4  22173
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