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Theorem ismtyhmeolem 26528
Description: Lemma for ismtyhmeo 26529. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
ismtyhmeo.1  |-  J  =  ( MetOpen `  M )
ismtyhmeo.2  |-  K  =  ( MetOpen `  N )
ismtyhmeolem.3  |-  ( ph  ->  M  e.  ( * Met `  X ) )
ismtyhmeolem.4  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
ismtyhmeolem.5  |-  ( ph  ->  F  e.  ( M 
Ismty  N ) )
Assertion
Ref Expression
ismtyhmeolem  |-  ( ph  ->  F  e.  ( J  Cn  K ) )

Proof of Theorem ismtyhmeolem
Dummy variables  u  r  w  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismtyhmeolem.5 . . . . 5  |-  ( ph  ->  F  e.  ( M 
Ismty  N ) )
2 ismtyhmeolem.3 . . . . . 6  |-  ( ph  ->  M  e.  ( * Met `  X ) )
3 ismtyhmeolem.4 . . . . . 6  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
4 isismty 26525 . . . . . 6  |-  ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  <->  ( F : X -1-1-onto-> Y  /\  A. x  e.  X  A. y  e.  X  ( x M y )  =  ( ( F `  x ) N ( F `  y ) ) ) ) )
52, 3, 4syl2anc 642 . . . . 5  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  <->  ( F : X -1-1-onto-> Y  /\  A. x  e.  X  A. y  e.  X  (
x M y )  =  ( ( F `
 x ) N ( F `  y
) ) ) ) )
61, 5mpbid 201 . . . 4  |-  ( ph  ->  ( F : X -1-1-onto-> Y  /\  A. x  e.  X  A. y  e.  X  ( x M y )  =  ( ( F `  x ) N ( F `  y ) ) ) )
76simpld 445 . . 3  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1of 5472 . . 3  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
97, 8syl 15 . 2  |-  ( ph  ->  F : X --> Y )
103adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  N  e.  ( * Met `  Y ) )
112adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  M  e.  ( * Met `  X ) )
12 ismtycnv 26526 . . . . . . . . . 10  |-  ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
132, 3, 12syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
141, 13mpd 14 . . . . . . . 8  |-  ( ph  ->  `' F  e.  ( N  Ismty  M ) )
1514adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  `' F  e.  ( N  Ismty  M ) )
16 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  w  e.  Y )
17 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
r  e.  RR* )
18 ismtyima 26527 . . . . . . 7  |-  ( ( ( N  e.  ( * Met `  Y
)  /\  M  e.  ( * Met `  X
)  /\  `' F  e.  ( N  Ismty  M ) )  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
1910, 11, 15, 16, 17, 18syl32anc 1190 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
20 f1ocnv 5485 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
21 f1of 5472 . . . . . . . . 9  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
227, 20, 213syl 18 . . . . . . . 8  |-  ( ph  ->  `' F : Y --> X )
23 simpl 443 . . . . . . . 8  |-  ( ( w  e.  Y  /\  r  e.  RR* )  ->  w  e.  Y )
24 ffvelrn 5663 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  w  e.  Y )  ->  ( `' F `  w )  e.  X
)
2522, 23, 24syl2an 463 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F `  w )  e.  X
)
26 ismtyhmeo.1 . . . . . . . 8  |-  J  =  ( MetOpen `  M )
2726blopn 18046 . . . . . . 7  |-  ( ( M  e.  ( * Met `  X )  /\  ( `' F `  w )  e.  X  /\  r  e.  RR* )  ->  ( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2811, 25, 17, 27syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2919, 28eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  e.  J )
3029ralrimivva 2635 . . . 4  |-  ( ph  ->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
31 fveq2 5525 . . . . . . . 8  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( (
ball `  N ) `  <. w ,  r
>. ) )
32 df-ov 5861 . . . . . . . 8  |-  ( w ( ball `  N
) r )  =  ( ( ball `  N
) `  <. w ,  r >. )
3331, 32syl6eqr 2333 . . . . . . 7  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( w ( ball `  N
) r ) )
3433imaeq2d 5012 . . . . . 6  |-  ( z  =  <. w ,  r
>.  ->  ( `' F " ( ( ball `  N
) `  z )
)  =  ( `' F " ( w ( ball `  N
) r ) ) )
3534eleq1d 2349 . . . . 5  |-  ( z  =  <. w ,  r
>.  ->  ( ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  ( `' F " ( w ( ball `  N ) r ) )  e.  J ) )
3635ralxp 4827 . . . 4  |-  ( A. z  e.  ( Y  X.  RR* ) ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
3730, 36sylibr 203 . . 3  |-  ( ph  ->  A. z  e.  ( Y  X.  RR* )
( `' F "
( ( ball `  N
) `  z )
)  e.  J )
38 blf 17961 . . . . 5  |-  ( N  e.  ( * Met `  Y )  ->  ( ball `  N ) : ( Y  X.  RR* )
--> ~P Y )
393, 38syl 15 . . . 4  |-  ( ph  ->  ( ball `  N
) : ( Y  X.  RR* ) --> ~P Y
)
40 ffn 5389 . . . 4  |-  ( (
ball `  N ) : ( Y  X.  RR* ) --> ~P Y  -> 
( ball `  N )  Fn  ( Y  X.  RR* ) )
41 imaeq2 5008 . . . . . 6  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( `' F " u )  =  ( `' F "
( ( ball `  N
) `  z )
) )
4241eleq1d 2349 . . . . 5  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( ( `' F " u )  e.  J  <->  ( `' F " ( ( ball `  N ) `  z
) )  e.  J
) )
4342ralrn 5668 . . . 4  |-  ( (
ball `  N )  Fn  ( Y  X.  RR* )  ->  ( A. u  e.  ran  ( ball `  N
) ( `' F " u )  e.  J  <->  A. z  e.  ( Y  X.  RR* ) ( `' F " ( (
ball `  N ) `  z ) )  e.  J ) )
4439, 40, 433syl 18 . . 3  |-  ( ph  ->  ( A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J  <->  A. z  e.  ( Y  X.  RR* ) ( `' F " ( (
ball `  N ) `  z ) )  e.  J ) )
4537, 44mpbird 223 . 2  |-  ( ph  ->  A. u  e.  ran  ( ball `  N )
( `' F "
u )  e.  J
)
4626mopntopon 17985 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
472, 46syl 15 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
48 ismtyhmeo.2 . . . . 5  |-  K  =  ( MetOpen `  N )
4948mopnval 17984 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( topGen `  ran  ( ball `  N )
) )
503, 49syl 15 . . 3  |-  ( ph  ->  K  =  ( topGen ` 
ran  ( ball `  N
) ) )
5148mopntopon 17985 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  e.  (TopOn `  Y )
)
523, 51syl 15 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5347, 50, 52tgcn 16982 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J
) ) )
549, 45, 53mpbir2and 888 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ~Pcpw 3625   <.cop 3643    X. cxp 4687   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372  TopOnctopon 16632    Cn ccn 16954    Ismty cismty 26522
This theorem is referenced by:  ismtyhmeo  26529
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-map 6774  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-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-ismty 26523
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