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Theorem fmfnfmlem3 20192
Description: Lemma for fmfnfm 20194. (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 20090 . . . . . . . . 9  |-  ( ( L  e.  ( Fil `  X )  /\  y  e.  L  /\  z  e.  L )  ->  (
y  i^i  z )  e.  L )
323expb 1197 . . . . . . . 8  |-  ( ( L  e.  ( Fil `  X )  /\  (
y  e.  L  /\  z  e.  L )
)  ->  ( y  i^i  z )  e.  L
)
41, 3sylan 471 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( y  i^i  z
)  e.  L )
5 fmfnfm.f . . . . . . . . 9  |-  ( ph  ->  F : Y --> X )
6 ffun 5731 . . . . . . . . 9  |-  ( F : Y --> X  ->  Fun  F )
7 funcnvcnv 5644 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
8 imain 5662 . . . . . . . . . 10  |-  ( Fun  `' `' F  ->  ( `' F " ( y  i^i  z ) )  =  ( ( `' F " y )  i^i  ( `' F " z ) ) )
98eqcomd 2475 . . . . . . . . 9  |-  ( Fun  `' `' F  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
( y  i^i  z
) ) )
105, 6, 7, 94syl 21 . . . . . . . 8  |-  ( ph  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
1110adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
12 imaeq2 5331 . . . . . . . . 9  |-  ( x  =  ( y  i^i  z )  ->  ( `' F " x )  =  ( `' F " ( y  i^i  z
) ) )
1312eqeq2d 2481 . . . . . . . 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 3214 . . . . . . 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 661 . . . . . 6  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  ->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " x ) )
16 ineq12 3695 . . . . . . . 8  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( s  i^i  t )  =  ( ( `' F "
y )  i^i  ( `' F " z ) ) )
1716eqeq1d 2469 . . . . . . 7  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( (
s  i^i  t )  =  ( `' F " x )  <->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) ) )
1817rexbidv 2973 . . . . . 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 222 . . . . 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 2962 . . . 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 5331 . . . . . . . 8  |-  ( x  =  y  ->  ( `' F " x )  =  ( `' F " y ) )
2221eqeq2d 2481 . . . . . . 7  |-  ( x  =  y  ->  (
s  =  ( `' F " x )  <-> 
s  =  ( `' F " y ) ) )
2322cbvrexv 3089 . . . . . 6  |-  ( E. x  e.  L  s  =  ( `' F " x )  <->  E. y  e.  L  s  =  ( `' F " y ) )
24 imaeq2 5331 . . . . . . . 8  |-  ( x  =  z  ->  ( `' F " x )  =  ( `' F " z ) )
2524eqeq2d 2481 . . . . . . 7  |-  ( x  =  z  ->  (
t  =  ( `' F " x )  <-> 
t  =  ( `' F " z ) ) )
2625cbvrexv 3089 . . . . . 6  |-  ( E. x  e.  L  t  =  ( `' F " x )  <->  E. z  e.  L  t  =  ( `' F " z ) )
2723, 26anbi12i 697 . . . . 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 3116 . . . . . . 7  |-  s  e. 
_V
29 eqid 2467 . . . . . . . 8  |-  ( x  e.  L  |->  ( `' F " x ) )  =  ( x  e.  L  |->  ( `' F " x ) )
3029elrnmpt 5247 . . . . . . 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 3116 . . . . . . 7  |-  t  e. 
_V
3329elrnmpt 5247 . . . . . . 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 697 . . . . 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 3029 . . . . 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 277 . . . 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 4588 . . . . 5  |-  ( s  i^i  t )  e. 
_V
3929elrnmpt 5247 . . . . 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 270 . . 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 2884 . 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 6128 . . 3  |-  ( L  e.  ( Fil `  X
)  ->  ( x  e.  L  |->  ( `' F " x ) )  e.  _V )
44 rnexg 6713 . . 3  |-  ( ( x  e.  L  |->  ( `' F " x ) )  e.  _V  ->  ran  ( x  e.  L  |->  ( `' F "
x ) )  e. 
_V )
45 inficl 7881 . . 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 21 . 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 210 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   ficfi 7866   fBascfbas 18177   Filcfil 20081    FilMap cfm 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-fbas 18187  df-fil 20082
This theorem is referenced by:  fmfnfmlem4  20193
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