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Theorem ismtyhmeolem 31546
Description: Lemma for ismtyhmeo 31547. (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 31543 . . . . . 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 659 . . . . 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 457 . . 3  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1of 5753 . . 3  |-  ( F : X -1-1-onto-> Y  ->  F : X
--> Y )
97, 8syl 17 . 2  |-  ( ph  ->  F : X --> Y )
103adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  N  e.  ( *Met `  Y ) )
112adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  ->  M  e.  ( *Met `  X ) )
12 ismtycnv 31544 . . . . . . . . . 10  |-  ( ( M  e.  ( *Met `  X )  /\  N  e.  ( *Met `  Y
) )  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
132, 3, 12syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  ( M  Ismty  N )  ->  `' F  e.  ( N  Ismty  M ) ) )
141, 13mpd 15 . . . . . . . 8  |-  ( ph  ->  `' F  e.  ( N  Ismty  M ) )
1514adantr 463 . . . . . . 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 31545 . . . . . . 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 1236 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  =  ( ( `' F `  w ) ( ball `  M
) r ) )
20 f1ocnv 5765 . . . . . . . . 9  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
21 f1of 5753 . . . . . . . . 9  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
227, 20, 213syl 20 . . . . . . . 8  |-  ( ph  ->  `' F : Y --> X )
23 simpl 455 . . . . . . . 8  |-  ( ( w  e.  Y  /\  r  e.  RR* )  ->  w  e.  Y )
24 ffvelrn 5961 . . . . . . . 8  |-  ( ( `' F : Y --> X  /\  w  e.  Y )  ->  ( `' F `  w )  e.  X
)
2522, 23, 24syl2an 475 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F `  w )  e.  X
)
26 ismtyhmeo.1 . . . . . . . 8  |-  J  =  ( MetOpen `  M )
2726blopn 21185 . . . . . . 7  |-  ( ( M  e.  ( *Met `  X )  /\  ( `' F `  w )  e.  X  /\  r  e.  RR* )  ->  ( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2811, 25, 17, 27syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( ( `' F `  w ) ( ball `  M ) r )  e.  J )
2919, 28eqeltrd 2488 . . . . 5  |-  ( (
ph  /\  ( w  e.  Y  /\  r  e.  RR* ) )  -> 
( `' F "
( w ( ball `  N ) r ) )  e.  J )
3029ralrimivva 2822 . . . 4  |-  ( ph  ->  A. w  e.  Y  A. r  e.  RR*  ( `' F " ( w ( ball `  N
) r ) )  e.  J )
31 fveq2 5803 . . . . . . . 8  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( (
ball `  N ) `  <. w ,  r
>. ) )
32 df-ov 6235 . . . . . . . 8  |-  ( w ( ball `  N
) r )  =  ( ( ball `  N
) `  <. w ,  r >. )
3331, 32syl6eqr 2459 . . . . . . 7  |-  ( z  =  <. w ,  r
>.  ->  ( ( ball `  N ) `  z
)  =  ( w ( ball `  N
) r ) )
3433imaeq2d 5276 . . . . . 6  |-  ( z  =  <. w ,  r
>.  ->  ( `' F " ( ( ball `  N
) `  z )
)  =  ( `' F " ( w ( ball `  N
) r ) ) )
3534eleq1d 2469 . . . . 5  |-  ( z  =  <. w ,  r
>.  ->  ( ( `' F " ( (
ball `  N ) `  z ) )  e.  J  <->  ( `' F " ( w ( ball `  N ) r ) )  e.  J ) )
3635ralxp 5084 . . . 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 21092 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  ( ball `  N ) : ( Y  X.  RR* )
--> ~P Y )
39 ffn 5668 . . . 4  |-  ( (
ball `  N ) : ( Y  X.  RR* ) --> ~P Y  -> 
( ball `  N )  Fn  ( Y  X.  RR* ) )
40 imaeq2 5272 . . . . . 6  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( `' F " u )  =  ( `' F "
( ( ball `  N
) `  z )
) )
4140eleq1d 2469 . . . . 5  |-  ( u  =  ( ( ball `  N ) `  z
)  ->  ( ( `' F " u )  e.  J  <->  ( `' F " ( ( ball `  N ) `  z
) )  e.  J
) )
4241ralrn 5966 . . . 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 21124 . . . 4  |-  ( M  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
)
462, 45syl 17 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
47 ismtyhmeo.2 . . . . 5  |-  K  =  ( MetOpen `  N )
4847mopnval 21123 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  =  ( topGen `  ran  ( ball `  N )
) )
493, 48syl 17 . . 3  |-  ( ph  ->  K  =  ( topGen ` 
ran  ( ball `  N
) ) )
5047mopntopon 21124 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  e.  (TopOn `  Y )
)
513, 50syl 17 . . 3  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5246, 49, 51tgcn 19936 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. u  e. 
ran  ( ball `  N
) ( `' F " u )  e.  J
) ) )
539, 44, 52mpbir2and 921 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   ~Pcpw 3952   <.cop 3975    X. cxp 4938   `'ccnv 4939   ran crn 4941   "cima 4943    Fn wfn 5518   -->wf 5519   -1-1-onto->wf1o 5522   ` cfv 5523  (class class class)co 6232   RR*cxr 9575   topGenctg 14942   *Metcxmt 18613   ballcbl 18615   MetOpencmopn 18618  TopOnctopon 19577    Cn ccn 19908    Ismty cismty 31540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-topgen 14948  df-psmet 18621  df-xmet 18622  df-bl 18624  df-mopn 18625  df-top 19581  df-bases 19583  df-topon 19584  df-cn 19911  df-ismty 31541
This theorem is referenced by:  ismtyhmeo  31547
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