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Theorem ulmdvlem2 20317
Description: Lemma for ulmdv 20319. (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 6106 . . . . . . 7  |-  ( S  _D  ( F `  k ) )  e. 
_V
21rgenw 2773 . . . . . 6  |-  A. k  e.  Z  ( S  _D  ( F `  k
) )  e.  _V
3 eqid 2436 . . . . . . 7  |-  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )  =  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )
43fnmpt 5571 . . . . . 6  |-  ( A. k  e.  Z  ( S  _D  ( F `  k ) )  e. 
_V  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )  Fn  Z
)
52, 4mp1i 12 . . . . 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 20300 . . . . 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 643 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( S  _D  ( F `  k )
) ) : Z --> ( CC  ^m  X ) )
93fmpt 5890 . . . 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 204 . . 3  |-  ( ph  ->  A. k  e.  Z  ( S  _D  ( F `  k )
)  e.  ( CC 
^m  X ) )
1110r19.21bi 2804 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( S  _D  ( F `  k ) )  e.  ( CC  ^m  X
) )
12 elmapi 7038 . 2  |-  ( ( S  _D  ( F `
 k ) )  e.  ( CC  ^m  X )  ->  ( S  _D  ( F `  k ) ) : X --> CC )
13 fdm 5595 . 2  |-  ( ( S  _D  ( F `
 k ) ) : X --> CC  ->  dom  ( S  _D  ( F `  k )
)  =  X )
1411, 12, 133syl 19 1  |-  ( (
ph  /\  k  e.  Z )  ->  dom  ( S  _D  ( F `  k )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   {cpr 3815   class class class wbr 4212    e. cmpt 4266   dom cdm 4878    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988   RRcr 8989   ZZcz 10282   ZZ>=cuz 10488    ~~> cli 12278    _D cdv 19750   ~~> uculm 20292
This theorem is referenced by:  ulmdvlem3  20318  ulmdv  20319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-pm 7021  df-neg 9294  df-z 10283  df-uz 10489  df-ulm 20293
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