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Theorem ulmdvlem2 19778
Description: Lemma for ulmdv 19780. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
ulmdv.z  |-  Z  =  ( ZZ>= `  M )
ulmdv.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
ulmdv.m  |-  ( ph  ->  M  e.  ZZ )
ulmdv.f  |-  ( ph  ->  F : Z --> ( CC 
^m  X ) )
ulmdv.g  |-  ( ph  ->  G : X --> CC )
ulmdv.l  |-  ( (
ph  /\  z  e.  X )  ->  (
k  e.  Z  |->  ( ( F `  k
) `  z )
)  ~~>  ( G `  z ) )
ulmdv.u  |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) ) ( ~~> u `  X ) H )
Assertion
Ref Expression
ulmdvlem2  |-  ( (
ph  /\  k  e.  Z )  ->  dom  ( S  _D  ( F `  k )
)  =  X )
Distinct variable groups:    z, k, F    z, G    z, H    k, M    ph, k, z    S, k, z    k, X, z   
k, Z, z
Allowed substitution hints:    G( k)    H( k)    M( z)

Proof of Theorem ulmdvlem2
StepHypRef Expression
1 ovex 5883 . . . . . . 7  |-  ( S  _D  ( F `  k ) )  e. 
_V
21rgenw 2610 . . . . . 6  |-  A. k  e.  Z  ( S  _D  ( F `  k
) )  e.  _V
3 eqid 2283 . . . . . . 7  |-  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )  =  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )
43fnmpt 5370 . . . . . 6  |-  ( A. k  e.  Z  ( S  _D  ( F `  k ) )  e. 
_V  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )  Fn  Z
)
52, 4mp1i 11 . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) )  Fn  Z
)
6 ulmdv.u . . . . 5  |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) ) ( ~~> u `  X ) H )
7 ulmf2 19763 . . . . 5  |-  ( ( ( k  e.  Z  |->  ( S  _D  ( F `  k )
) )  Fn  Z  /\  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) ) ( ~~> u `  X ) H )  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) : Z --> ( CC  ^m  X ) )
85, 6, 7syl2anc 642 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) ) : Z --> ( CC  ^m  X ) )
93fmpt 5681 . . . 4  |-  ( A. k  e.  Z  ( S  _D  ( F `  k ) )  e.  ( CC  ^m  X
)  <->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) ) : Z --> ( CC  ^m  X ) )
108, 9sylibr 203 . . 3  |-  ( ph  ->  A. k  e.  Z  ( S  _D  ( F `  k )
)  e.  ( CC 
^m  X ) )
1110r19.21bi 2641 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( S  _D  ( F `  k ) )  e.  ( CC  ^m  X
) )
12 elmapi 6792 . 2  |-  ( ( S  _D  ( F `
 k ) )  e.  ( CC  ^m  X )  ->  ( S  _D  ( F `  k ) ) : X --> CC )
13 fdm 5393 . 2  |-  ( ( S  _D  ( F `
 k ) ) : X --> CC  ->  dom  ( S  _D  ( F `  k )
)  =  X )
1411, 12, 133syl 18 1  |-  ( (
ph  /\  k  e.  Z )  ->  dom  ( S  _D  ( F `  k )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   {cpr 3641   class class class wbr 4023    e. cmpt 4077   dom cdm 4689    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   RRcr 8736   ZZcz 10024   ZZ>=cuz 10230    ~~> cli 11958    _D cdv 19213   ~~> uculm 19755
This theorem is referenced by:  ulmdvlem3  19779  ulmdv  19780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-pm 6775  df-neg 9040  df-z 10025  df-uz 10231  df-ulm 19756
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