Metamath Proof Explorer


Theorem ulmdvlem2

Description: Lemma for ulmdv . (Contributed by Mario Carneiro, 8-May-2015)

Ref Expression
Hypotheses 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 ulmdvlem2
|- ( ( ph /\ k e. Z ) -> dom ( S _D ( F ` k ) ) = X )

Proof

Step Hyp Ref Expression
1 ulmdv.z
 |-  Z = ( ZZ>= ` M )
2 ulmdv.s
 |-  ( ph -> S e. { RR , CC } )
3 ulmdv.m
 |-  ( ph -> M e. ZZ )
4 ulmdv.f
 |-  ( ph -> F : Z --> ( CC ^m X ) )
5 ulmdv.g
 |-  ( ph -> G : X --> CC )
6 ulmdv.l
 |-  ( ( ph /\ z e. X ) -> ( k e. Z |-> ( ( F ` k ) ` z ) ) ~~> ( G ` z ) )
7 ulmdv.u
 |-  ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) ( ~~>u ` X ) H )
8 ovex
 |-  ( S _D ( F ` k ) ) e. _V
9 8 rgenw
 |-  A. k e. Z ( S _D ( F ` k ) ) e. _V
10 eqid
 |-  ( k e. Z |-> ( S _D ( F ` k ) ) ) = ( k e. Z |-> ( S _D ( F ` k ) ) )
11 10 fnmpt
 |-  ( A. k e. Z ( S _D ( F ` k ) ) e. _V -> ( k e. Z |-> ( S _D ( F ` k ) ) ) Fn Z )
12 9 11 mp1i
 |-  ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) Fn Z )
13 ulmf2
 |-  ( ( ( 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 ) )
14 12 7 13 syl2anc
 |-  ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) : Z --> ( CC ^m X ) )
15 14 fvmptelrn
 |-  ( ( ph /\ k e. Z ) -> ( S _D ( F ` k ) ) e. ( CC ^m X ) )
16 elmapi
 |-  ( ( S _D ( F ` k ) ) e. ( CC ^m X ) -> ( S _D ( F ` k ) ) : X --> CC )
17 fdm
 |-  ( ( S _D ( F ` k ) ) : X --> CC -> dom ( S _D ( F ` k ) ) = X )
18 15 16 17 3syl
 |-  ( ( ph /\ k e. Z ) -> dom ( S _D ( F ` k ) ) = X )