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Theorem ulmdvlem2 19831
Description: Lemma for ulmdv 19833. (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 5925 . . . . . . 7  |-  ( S  _D  ( F `  k ) )  e. 
_V
21rgenw 2644 . . . . . 6  |-  A. k  e.  Z  ( S  _D  ( F `  k
) )  e.  _V
3 eqid 2316 . . . . . . 7  |-  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )  =  ( k  e.  Z  |->  ( S  _D  ( F `  k ) ) )
43fnmpt 5407 . . . . . 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 19816 . . . . 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 5719 . . . 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 2675 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( S  _D  ( F `  k ) )  e.  ( CC  ^m  X
) )
12 elmapi 6835 . 2  |-  ( ( S  _D  ( F `
 k ) )  e.  ( CC  ^m  X )  ->  ( S  _D  ( F `  k ) ) : X --> CC )
13 fdm 5431 . 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 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822   {cpr 3675   class class class wbr 4060    e. cmpt 4114   dom cdm 4726    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    ^m cmap 6815   CCcc 8780   RRcr 8781   ZZcz 10071   ZZ>=cuz 10277    ~~> cli 12005    _D cdv 19266   ~~> uculm 19808
This theorem is referenced by:  ulmdvlem3  19832  ulmdv  19833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-map 6817  df-pm 6818  df-neg 9085  df-z 10072  df-uz 10278  df-ulm 19809
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