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Theorem ismtyhmeolem 28708
Description: Lemma for ismtyhmeo 28709. (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 28705 . . . . . 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 661 . . . . 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 210 . . . 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 459 . . 3  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1of 5646 . . 3  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
97, 8syl 16 . 2  |-  ( ph  ->  F : X --> Y )
103adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  N  e.  ( *Met `  Y ) )
112adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  M  e.  ( *Met `  X ) )
12 ismtycnv 28706 . . . . . . . . . 10  |-  ( ( M  e.  ( *Met `  X )  /\  N  e.  ( *Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
132, 3, 12syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
141, 13mpd 15 . . . . . . . 8  |-  ( ph  ->  `' F  e.  ( N  Ismty  M ) )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  `' F  e.  ( N  Ismty  M ) )
16 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  w  e.  Y )
17 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
r  e.  RR* )
18 ismtyima 28707 . . . . . . 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 1226 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
20 f1ocnv 5658 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
21 f1of 5646 . . . . . . . . 9  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
227, 20, 213syl 20 . . . . . . . 8  |-  ( ph  ->  `' F : Y --> X )
23 simpl 457 . . . . . . . 8  |-  ( ( w  e.  Y  /\  r  e.  RR* )  ->  w  e.  Y )
24 ffvelrn 5846 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  w  e.  Y )  ->  ( `' F `  w )  e.  X
)
2522, 23, 24syl2an 477 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F `  w )  e.  X
)
26 ismtyhmeo.1 . . . . . . . 8  |-  J  =  ( MetOpen `  M )
2726blopn 20080 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( `' F `  w )  e.  X  /\  r  e.  RR* )  ->  ( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2811, 25, 17, 27syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2919, 28eqeltrd 2517 . . . . 5  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  e.  J )
3029ralrimivva 2813 . . . 4  |-  ( ph  ->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
31 fveq2 5696 . . . . . . . 8  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( (
ball `  N ) `  <. w ,  r
>. ) )
32 df-ov 6099 . . . . . . . 8  |-  ( w ( ball `  N
) r )  =  ( ( ball `  N
) `  <. w ,  r >. )
3331, 32syl6eqr 2493 . . . . . . 7  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( w ( ball `  N
) r ) )
3433imaeq2d 5174 . . . . . 6  |-  ( z  =  <. w ,  r
>.  ->  ( `' F " ( ( ball `  N
) `  z )
)  =  ( `' F " ( w ( ball `  N
) r ) ) )
3534eleq1d 2509 . . . . 5  |-  ( z  =  <. w ,  r
>.  ->  ( ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  ( `' F " ( w ( ball `  N ) r ) )  e.  J ) )
3635ralxp 4986 . . . 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 212 . . 3  |-  ( ph  ->  A. z  e.  ( Y  X.  RR* )
( `' F "
( ( ball `  N
) `  z )
)  e.  J )
38 blf 19987 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  ( ball `  N ) : ( Y  X.  RR* )
--> ~P Y )
39 ffn 5564 . . . 4  |-  ( (
ball `  N ) : ( Y  X.  RR* ) --> ~P Y  -> 
( ball `  N )  Fn  ( Y  X.  RR* ) )
40 imaeq2 5170 . . . . . 6  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( `' F " u )  =  ( `' F "
( ( ball `  N
) `  z )
) )
4140eleq1d 2509 . . . . 5  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( ( `' F " u )  e.  J  <->  ( `' F " ( ( ball `  N ) `  z
) )  e.  J
) )
4241ralrn 5851 . . . 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 ) )
433, 38, 39, 424syl 21 . . 3  |-  ( ph  ->  ( A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J  <->  A. z  e.  ( Y  X.  RR* ) ( `' F " ( (
ball `  N ) `  z ) )  e.  J ) )
4437, 43mpbird 232 . 2  |-  ( ph  ->  A. u  e.  ran  ( ball `  N )
( `' F "
u )  e.  J
)
4526mopntopon 20019 . . . 4  |-  ( M  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
)
462, 45syl 16 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
47 ismtyhmeo.2 . . . . 5  |-  K  =  ( MetOpen `  N )
4847mopnval 20018 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  =  ( topGen `  ran  ( ball `  N )
) )
493, 48syl 16 . . 3  |-  ( ph  ->  K  =  ( topGen ` 
ran  ( ball `  N
) ) )
5047mopntopon 20019 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  e.  (TopOn `  Y )
)
513, 50syl 16 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5246, 49, 51tgcn 18861 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J
) ) )
539, 44, 52mpbir2and 913 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   ~Pcpw 3865   <.cop 3888    X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5418   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   RR*cxr 9422   topGenctg 14381   *Metcxmt 17806   ballcbl 17808   MetOpencmopn 17811  TopOnctopon 18504    Cn ccn 18833    Ismty cismty 28702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-cn 18836  df-ismty 28703
This theorem is referenced by:  ismtyhmeo  28709
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