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Theorem ismtyhmeolem 26411
Description: Lemma for ismtyhmeo 26412. (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 26408 . . . . . 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 643 . . . . 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 202 . . . 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 446 . . 3  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1of 5641 . . 3  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
97, 8syl 16 . 2  |-  ( ph  ->  F : X --> Y )
103adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  N  e.  ( * Met `  Y ) )
112adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  M  e.  ( * Met `  X ) )
12 ismtycnv 26409 . . . . . . . . . 10  |-  ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
132, 3, 12syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
141, 13mpd 15 . . . . . . . 8  |-  ( ph  ->  `' F  e.  ( N  Ismty  M ) )
1514adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  `' F  e.  ( N  Ismty  M ) )
16 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  w  e.  Y )
17 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
r  e.  RR* )
18 ismtyima 26410 . . . . . . 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 1192 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
20 f1ocnv 5654 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
21 f1of 5641 . . . . . . . . 9  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
227, 20, 213syl 19 . . . . . . . 8  |-  ( ph  ->  `' F : Y --> X )
23 simpl 444 . . . . . . . 8  |-  ( ( w  e.  Y  /\  r  e.  RR* )  ->  w  e.  Y )
24 ffvelrn 5835 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  w  e.  Y )  ->  ( `' F `  w )  e.  X
)
2522, 23, 24syl2an 464 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F `  w )  e.  X
)
26 ismtyhmeo.1 . . . . . . . 8  |-  J  =  ( MetOpen `  M )
2726blopn 18491 . . . . . . 7  |-  ( ( M  e.  ( * Met `  X )  /\  ( `' F `  w )  e.  X  /\  r  e.  RR* )  ->  ( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2811, 25, 17, 27syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2919, 28eqeltrd 2486 . . . . 5  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  e.  J )
3029ralrimivva 2766 . . . 4  |-  ( ph  ->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
31 fveq2 5695 . . . . . . . 8  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( (
ball `  N ) `  <. w ,  r
>. ) )
32 df-ov 6051 . . . . . . . 8  |-  ( w ( ball `  N
) r )  =  ( ( ball `  N
) `  <. w ,  r >. )
3331, 32syl6eqr 2462 . . . . . . 7  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( w ( ball `  N
) r ) )
3433imaeq2d 5170 . . . . . 6  |-  ( z  =  <. w ,  r
>.  ->  ( `' F " ( ( ball `  N
) `  z )
)  =  ( `' F " ( w ( ball `  N
) r ) ) )
3534eleq1d 2478 . . . . 5  |-  ( z  =  <. w ,  r
>.  ->  ( ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  ( `' F " ( w ( ball `  N ) r ) )  e.  J ) )
3635ralxp 4983 . . . 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 204 . . 3  |-  ( ph  ->  A. z  e.  ( Y  X.  RR* )
( `' F "
( ( ball `  N
) `  z )
)  e.  J )
38 blf 18398 . . . . 5  |-  ( N  e.  ( * Met `  Y )  ->  ( ball `  N ) : ( Y  X.  RR* )
--> ~P Y )
393, 38syl 16 . . . 4  |-  ( ph  ->  ( ball `  N
) : ( Y  X.  RR* ) --> ~P Y
)
40 ffn 5558 . . . 4  |-  ( (
ball `  N ) : ( Y  X.  RR* ) --> ~P Y  -> 
( ball `  N )  Fn  ( Y  X.  RR* ) )
41 imaeq2 5166 . . . . . 6  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( `' F " u )  =  ( `' F "
( ( ball `  N
) `  z )
) )
4241eleq1d 2478 . . . . 5  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( ( `' F " u )  e.  J  <->  ( `' F " ( ( ball `  N ) `  z
) )  e.  J
) )
4342ralrn 5840 . . . 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 19 . . 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 224 . 2  |-  ( ph  ->  A. u  e.  ran  ( ball `  N )
( `' F "
u )  e.  J
)
4626mopntopon 18430 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
472, 46syl 16 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
48 ismtyhmeo.2 . . . . 5  |-  K  =  ( MetOpen `  N )
4948mopnval 18429 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( topGen `  ran  ( ball `  N )
) )
503, 49syl 16 . . 3  |-  ( ph  ->  K  =  ( topGen ` 
ran  ( ball `  N
) ) )
5148mopntopon 18430 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  e.  (TopOn `  Y )
)
523, 51syl 16 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5347, 50, 52tgcn 17278 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J
) ) )
549, 45, 53mpbir2and 889 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   ~Pcpw 3767   <.cop 3785    X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5416   -->wf 5417   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   RR*cxr 9083   topGenctg 13628   * Metcxmt 16649   ballcbl 16651   MetOpencmopn 16654  TopOnctopon 16922    Cn ccn 17250    Ismty cismty 26405
This theorem is referenced by:  ismtyhmeo  26412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-cn 17253  df-ismty 26406
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