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Theorem ismtyhmeolem 26527
Description: Lemma for ismtyhmeo 26528. (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 26524 . . . . . 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 644 . . . . 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 203 . . . 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 447 . . 3  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1of 5677 . . 3  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
97, 8syl 16 . 2  |-  ( ph  ->  F : X --> Y )
103adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  N  e.  ( * Met `  Y ) )
112adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  M  e.  ( * Met `  X ) )
12 ismtycnv 26525 . . . . . . . . . 10  |-  ( ( M  e.  ( * Met `  X )  /\  N  e.  ( * Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
132, 3, 12syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
141, 13mpd 15 . . . . . . . 8  |-  ( ph  ->  `' F  e.  ( N  Ismty  M ) )
1514adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  `' F  e.  ( N  Ismty  M ) )
16 simprl 734 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  w  e.  Y )
17 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
r  e.  RR* )
18 ismtyima 26526 . . . . . . 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 1193 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
20 f1ocnv 5690 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
21 f1of 5677 . . . . . . . . 9  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
227, 20, 213syl 19 . . . . . . . 8  |-  ( ph  ->  `' F : Y --> X )
23 simpl 445 . . . . . . . 8  |-  ( ( w  e.  Y  /\  r  e.  RR* )  ->  w  e.  Y )
24 ffvelrn 5871 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  w  e.  Y )  ->  ( `' F `  w )  e.  X
)
2522, 23, 24syl2an 465 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F `  w )  e.  X
)
26 ismtyhmeo.1 . . . . . . . 8  |-  J  =  ( MetOpen `  M )
2726blopn 18535 . . . . . . 7  |-  ( ( M  e.  ( * Met `  X )  /\  ( `' F `  w )  e.  X  /\  r  e.  RR* )  ->  ( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2811, 25, 17, 27syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2919, 28eqeltrd 2512 . . . . 5  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  e.  J )
3029ralrimivva 2800 . . . 4  |-  ( ph  ->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
31 fveq2 5731 . . . . . . . 8  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( (
ball `  N ) `  <. w ,  r
>. ) )
32 df-ov 6087 . . . . . . . 8  |-  ( w ( ball `  N
) r )  =  ( ( ball `  N
) `  <. w ,  r >. )
3331, 32syl6eqr 2488 . . . . . . 7  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( w ( ball `  N
) r ) )
3433imaeq2d 5206 . . . . . 6  |-  ( z  =  <. w ,  r
>.  ->  ( `' F " ( ( ball `  N
) `  z )
)  =  ( `' F " ( w ( ball `  N
) r ) ) )
3534eleq1d 2504 . . . . 5  |-  ( z  =  <. w ,  r
>.  ->  ( ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  ( `' F " ( w ( ball `  N ) r ) )  e.  J ) )
3635ralxp 5019 . . . 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 205 . . 3  |-  ( ph  ->  A. z  e.  ( Y  X.  RR* )
( `' F "
( ( ball `  N
) `  z )
)  e.  J )
38 blf 18442 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  ( ball `  N ) : ( Y  X.  RR* )
--> ~P Y )
39 ffn 5594 . . . 4  |-  ( (
ball `  N ) : ( Y  X.  RR* ) --> ~P Y  -> 
( ball `  N )  Fn  ( Y  X.  RR* ) )
40 imaeq2 5202 . . . . . 6  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( `' F " u )  =  ( `' F "
( ( ball `  N
) `  z )
) )
4140eleq1d 2504 . . . . 5  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( ( `' F " u )  e.  J  <->  ( `' F " ( ( ball `  N ) `  z
) )  e.  J
) )
4241ralrn 5876 . . . 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 20 . . 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 225 . 2  |-  ( ph  ->  A. u  e.  ran  ( ball `  N )
( `' F "
u )  e.  J
)
4526mopntopon 18474 . . . 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 18473 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( topGen `  ran  ( ball `  N )
) )
493, 48syl 16 . . 3  |-  ( ph  ->  K  =  ( topGen ` 
ran  ( ball `  N
) ) )
5047mopntopon 18474 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  e.  (TopOn `  Y )
)
513, 50syl 16 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5246, 49, 51tgcn 17321 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J
) ) )
539, 44, 52mpbir2and 890 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ~Pcpw 3801   <.cop 3819    X. cxp 4879   `'ccnv 4880   ran crn 4882   "cima 4884    Fn wfn 5452   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   RR*cxr 9124   topGenctg 13670   * Metcxmt 16691   ballcbl 16693   MetOpencmopn 16696  TopOnctopon 16964    Cn ccn 17293    Ismty cismty 26521
This theorem is referenced by:  ismtyhmeo  26528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-cn 17296  df-ismty 26522
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