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Theorem metcnpi2 20778
Description: Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 20775. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypotheses
Ref Expression
metcn.2  |-  J  =  ( MetOpen `  C )
metcn.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metcnpi2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( (
y C P )  <  x  ->  (
( F `  y
) D ( F `
 P ) )  <  A ) )
Distinct variable groups:    x, y, F    x, J, y    x, K, y    x, X, y   
x, Y, y    x, A, y    x, C, y   
x, D, y    x, P, y

Proof of Theorem metcnpi2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  F  e.  ( ( J  CnP  K ) `  P ) )
2 simpll 753 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  C  e.  ( *Met `  X
) )
3 simplr 754 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  D  e.  ( *Met `  Y
) )
4 eqid 2462 . . . . . . . . 9  |-  U. J  =  U. J
54cnprcl 19507 . . . . . . . 8  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  U. J )
65adantl 466 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  U. J )
7 metcn.2 . . . . . . . . 9  |-  J  =  ( MetOpen `  C )
87mopnuni 20674 . . . . . . . 8  |-  ( C  e.  ( *Met `  X )  ->  X  =  U. J )
98ad2antrr 725 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  X  =  U. J )
106, 9eleqtrrd 2553 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  X )
11 metcn.4 . . . . . . 7  |-  K  =  ( MetOpen `  D )
127, 11metcnp2 20775 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
z ) ) ) )
132, 3, 10, 12syl3anc 1223 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <  x  -> 
( ( F `  y ) D ( F `  P ) )  <  z ) ) ) )
141, 13mpbid 210 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
z ) ) )
1514simprd 463 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  z ) )
16 breq2 4446 . . . . . 6  |-  ( z  =  A  ->  (
( ( F `  y ) D ( F `  P ) )  <  z  <->  ( ( F `  y ) D ( F `  P ) )  < 
A ) )
1716imbi2d 316 . . . . 5  |-  ( z  =  A  ->  (
( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  z )  <-> 
( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  A ) ) )
1817rexralbidv 2976 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  RR+  A. y  e.  X  ( (
y C P )  <  x  ->  (
( F `  y
) D ( F `
 P ) )  <  z )  <->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
A ) ) )
1918rspccv 3206 . . 3  |-  ( A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
z )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
A ) ) )
2015, 19syl 16 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  < 
x  ->  ( ( F `  y ) D ( F `  P ) )  < 
A ) ) )
2120impr 619 1  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( (
y C P )  <  x  ->  (
( F `  y
) D ( F `
 P ) )  <  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   U.cuni 4240   class class class wbr 4442   -->wf 5577   ` cfv 5581  (class class class)co 6277    < clt 9619   RR+crp 11211   *Metcxmt 18169   MetOpencmopn 18174    CnP ccnp 19487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-topgen 14690  df-psmet 18177  df-xmet 18178  df-bl 18180  df-mopn 18181  df-top 19161  df-bases 19163  df-topon 19164  df-cnp 19490
This theorem is referenced by:  metcnpi3  20779  ftc1lem6  22172  ftc1cnnc  29655
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