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Theorem fmfnfmlem3 21020
Description: Lemma for fmfnfm 21022. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypotheses
Ref Expression
fmfnfm.b  |-  ( ph  ->  B  e.  ( fBas `  Y ) )
fmfnfm.l  |-  ( ph  ->  L  e.  ( Fil `  X ) )
fmfnfm.f  |-  ( ph  ->  F : Y --> X )
fmfnfm.fm  |-  ( ph  ->  ( ( X  FilMap  F ) `  B ) 
C_  L )
Assertion
Ref Expression
fmfnfmlem3  |-  ( ph  ->  ( fi `  ran  ( x  e.  L  |->  ( `' F "
x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) )
Distinct variable groups:    x, B    x, F    x, L    ph, x    x, X    x, Y

Proof of Theorem fmfnfmlem3
Dummy variables  s 
t  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmfnfm.l . . . . . . . 8  |-  ( ph  ->  L  e.  ( Fil `  X ) )
2 filin 20918 . . . . . . . . 9  |-  ( ( L  e.  ( Fil `  X )  /\  y  e.  L  /\  z  e.  L )  ->  (
y  i^i  z )  e.  L )
323expb 1216 . . . . . . . 8  |-  ( ( L  e.  ( Fil `  X )  /\  (
y  e.  L  /\  z  e.  L )
)  ->  ( y  i^i  z )  e.  L
)
41, 3sylan 478 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( y  i^i  z
)  e.  L )
5 fmfnfm.f . . . . . . . . 9  |-  ( ph  ->  F : Y --> X )
6 ffun 5754 . . . . . . . . 9  |-  ( F : Y --> X  ->  Fun  F )
7 funcnvcnv 5663 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
8 imain 5681 . . . . . . . . . 10  |-  ( Fun  `' `' F  ->  ( `' F " ( y  i^i  z ) )  =  ( ( `' F " y )  i^i  ( `' F " z ) ) )
98eqcomd 2468 . . . . . . . . 9  |-  ( Fun  `' `' F  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
( y  i^i  z
) ) )
105, 6, 7, 94syl 19 . . . . . . . 8  |-  ( ph  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
1110adantr 471 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
12 imaeq2 5183 . . . . . . . . 9  |-  ( x  =  ( y  i^i  z )  ->  ( `' F " x )  =  ( `' F " ( y  i^i  z
) ) )
1312eqeq2d 2472 . . . . . . . 8  |-  ( x  =  ( y  i^i  z )  ->  (
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " x )  <-> 
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) ) )
1413rspcev 3162 . . . . . . 7  |-  ( ( ( y  i^i  z
)  e.  L  /\  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )  ->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) )
154, 11, 14syl2anc 671 . . . . . 6  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  ->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " x ) )
16 ineq12 3641 . . . . . . . 8  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( s  i^i  t )  =  ( ( `' F "
y )  i^i  ( `' F " z ) ) )
1716eqeq1d 2464 . . . . . . 7  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( (
s  i^i  t )  =  ( `' F " x )  <->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) ) )
1817rexbidv 2913 . . . . . 6  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( E. x  e.  L  (
s  i^i  t )  =  ( `' F " x )  <->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) ) )
1915, 18syl5ibrcom 230 . . . . 5  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( s  =  ( `' F "
y )  /\  t  =  ( `' F " z ) )  ->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) ) )
2019rexlimdvva 2898 . . . 4  |-  ( ph  ->  ( E. y  e.  L  E. z  e.  L  ( s  =  ( `' F "
y )  /\  t  =  ( `' F " z ) )  ->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) ) )
21 imaeq2 5183 . . . . . . . 8  |-  ( x  =  y  ->  ( `' F " x )  =  ( `' F " y ) )
2221eqeq2d 2472 . . . . . . 7  |-  ( x  =  y  ->  (
s  =  ( `' F " x )  <-> 
s  =  ( `' F " y ) ) )
2322cbvrexv 3032 . . . . . 6  |-  ( E. x  e.  L  s  =  ( `' F " x )  <->  E. y  e.  L  s  =  ( `' F " y ) )
24 imaeq2 5183 . . . . . . . 8  |-  ( x  =  z  ->  ( `' F " x )  =  ( `' F " z ) )
2524eqeq2d 2472 . . . . . . 7  |-  ( x  =  z  ->  (
t  =  ( `' F " x )  <-> 
t  =  ( `' F " z ) ) )
2625cbvrexv 3032 . . . . . 6  |-  ( E. x  e.  L  t  =  ( `' F " x )  <->  E. z  e.  L  t  =  ( `' F " z ) )
2723, 26anbi12i 708 . . . . 5  |-  ( ( E. x  e.  L  s  =  ( `' F " x )  /\  E. x  e.  L  t  =  ( `' F " x ) )  <->  ( E. y  e.  L  s  =  ( `' F " y )  /\  E. z  e.  L  t  =  ( `' F " z ) ) )
28 vex 3060 . . . . . . 7  |-  s  e. 
_V
29 eqid 2462 . . . . . . . 8  |-  ( x  e.  L  |->  ( `' F " x ) )  =  ( x  e.  L  |->  ( `' F " x ) )
3029elrnmpt 5100 . . . . . . 7  |-  ( s  e.  _V  ->  (
s  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  s  =  ( `' F " x ) ) )
3128, 30ax-mp 5 . . . . . 6  |-  ( s  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  s  =  ( `' F " x ) )
32 vex 3060 . . . . . . 7  |-  t  e. 
_V
3329elrnmpt 5100 . . . . . . 7  |-  ( t  e.  _V  ->  (
t  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  t  =  ( `' F " x ) ) )
3432, 33ax-mp 5 . . . . . 6  |-  ( t  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  t  =  ( `' F " x ) )
3531, 34anbi12i 708 . . . . 5  |-  ( ( s  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  /\  t  e. 
ran  ( x  e.  L  |->  ( `' F " x ) ) )  <-> 
( E. x  e.  L  s  =  ( `' F " x )  /\  E. x  e.  L  t  =  ( `' F " x ) ) )
36 reeanv 2970 . . . . 5  |-  ( E. y  e.  L  E. z  e.  L  (
s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  <->  ( E. y  e.  L  s  =  ( `' F " y )  /\  E. z  e.  L  t  =  ( `' F " z ) ) )
3727, 35, 363bitr4i 285 . . . 4  |-  ( ( s  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  /\  t  e. 
ran  ( x  e.  L  |->  ( `' F " x ) ) )  <->  E. y  e.  L  E. z  e.  L  ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) ) )
3828inex1 4558 . . . . 5  |-  ( s  i^i  t )  e. 
_V
3929elrnmpt 5100 . . . . 5  |-  ( ( s  i^i  t )  e.  _V  ->  (
( s  i^i  t
)  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) ) )
4038, 39ax-mp 5 . . . 4  |-  ( ( s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) )
4120, 37, 403imtr4g 278 . . 3  |-  ( ph  ->  ( ( s  e. 
ran  ( x  e.  L  |->  ( `' F " x ) )  /\  t  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) )  ->  (
s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) ) )
4241ralrimivv 2820 . 2  |-  ( ph  ->  A. s  e.  ran  ( x  e.  L  |->  ( `' F "
x ) ) A. t  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) ( s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F "
x ) ) )
43 mptexg 6160 . . 3  |-  ( L  e.  ( Fil `  X
)  ->  ( x  e.  L  |->  ( `' F " x ) )  e.  _V )
44 rnexg 6752 . . 3  |-  ( ( x  e.  L  |->  ( `' F " x ) )  e.  _V  ->  ran  ( x  e.  L  |->  ( `' F "
x ) )  e. 
_V )
45 inficl 7965 . . 3  |-  ( ran  ( x  e.  L  |->  ( `' F "
x ) )  e. 
_V  ->  ( A. s  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) A. t  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) ( s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F "
x ) )  <->  ( fi ` 
ran  ( x  e.  L  |->  ( `' F " x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) ) )
461, 43, 44, 454syl 19 . 2  |-  ( ph  ->  ( A. s  e. 
ran  ( x  e.  L  |->  ( `' F " x ) ) A. t  e.  ran  ( x  e.  L  |->  ( `' F " x ) ) ( s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F "
x ) )  <->  ( fi ` 
ran  ( x  e.  L  |->  ( `' F " x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) ) )
4742, 46mpbid 215 1  |-  ( ph  ->  ( fi `  ran  ( x  e.  L  |->  ( `' F "
x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057    i^i cin 3415    C_ wss 3416    |-> cmpt 4475   `'ccnv 4852   ran crn 4854   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6315   ficfi 7950   fBascfbas 19007   Filcfil 20909    FilMap cfm 20997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-en 7596  df-fin 7599  df-fi 7951  df-fbas 19016  df-fil 20910
This theorem is referenced by:  fmfnfmlem4  21021
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