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Theorem fmfnfmlem3 20908
Description: Lemma for fmfnfm 20910. (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 20806 . . . . . . . . 9  |-  ( ( L  e.  ( Fil `  X )  /\  y  e.  L  /\  z  e.  L )  ->  (
y  i^i  z )  e.  L )
323expb 1206 . . . . . . . 8  |-  ( ( L  e.  ( Fil `  X )  /\  (
y  e.  L  /\  z  e.  L )
)  ->  ( y  i^i  z )  e.  L
)
41, 3sylan 473 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( y  i^i  z
)  e.  L )
5 fmfnfm.f . . . . . . . . 9  |-  ( ph  ->  F : Y --> X )
6 ffun 5686 . . . . . . . . 9  |-  ( F : Y --> X  ->  Fun  F )
7 funcnvcnv 5597 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
8 imain 5615 . . . . . . . . . 10  |-  ( Fun  `' `' F  ->  ( `' F " ( y  i^i  z ) )  =  ( ( `' F " y )  i^i  ( `' F " z ) ) )
98eqcomd 2429 . . . . . . . . 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 466 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
12 imaeq2 5121 . . . . . . . . 9  |-  ( x  =  ( y  i^i  z )  ->  ( `' F " x )  =  ( `' F " ( y  i^i  z
) ) )
1312eqeq2d 2433 . . . . . . . 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 3120 . . . . . . 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 665 . . . . . 6  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  ->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " x ) )
16 ineq12 3597 . . . . . . . 8  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( s  i^i  t )  =  ( ( `' F "
y )  i^i  ( `' F " z ) ) )
1716eqeq1d 2425 . . . . . . 7  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( (
s  i^i  t )  =  ( `' F " x )  <->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) ) )
1817rexbidv 2873 . . . . . 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 225 . . . . 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 2858 . . . 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 5121 . . . . . . . 8  |-  ( x  =  y  ->  ( `' F " x )  =  ( `' F " y ) )
2221eqeq2d 2433 . . . . . . 7  |-  ( x  =  y  ->  (
s  =  ( `' F " x )  <-> 
s  =  ( `' F " y ) ) )
2322cbvrexv 2992 . . . . . 6  |-  ( E. x  e.  L  s  =  ( `' F " x )  <->  E. y  e.  L  s  =  ( `' F " y ) )
24 imaeq2 5121 . . . . . . . 8  |-  ( x  =  z  ->  ( `' F " x )  =  ( `' F " z ) )
2524eqeq2d 2433 . . . . . . 7  |-  ( x  =  z  ->  (
t  =  ( `' F " x )  <-> 
t  =  ( `' F " z ) ) )
2625cbvrexv 2992 . . . . . 6  |-  ( E. x  e.  L  t  =  ( `' F " x )  <->  E. z  e.  L  t  =  ( `' F " z ) )
2723, 26anbi12i 701 . . . . 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 3020 . . . . . . 7  |-  s  e. 
_V
29 eqid 2423 . . . . . . . 8  |-  ( x  e.  L  |->  ( `' F " x ) )  =  ( x  e.  L  |->  ( `' F " x ) )
3029elrnmpt 5038 . . . . . . 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 3020 . . . . . . 7  |-  t  e. 
_V
3329elrnmpt 5038 . . . . . . 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 701 . . . . 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 2930 . . . . 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 280 . . . 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 4503 . . . . 5  |-  ( s  i^i  t )  e. 
_V
3929elrnmpt 5038 . . . . 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 273 . . 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 2780 . 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 6089 . . 3  |-  ( L  e.  ( Fil `  X
)  ->  ( x  e.  L  |->  ( `' F " x ) )  e.  _V )
44 rnexg 6678 . . 3  |-  ( ( x  e.  L  |->  ( `' F " x ) )  e.  _V  ->  ran  ( x  e.  L  |->  ( `' F "
x ) )  e. 
_V )
45 inficl 7887 . . 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 213 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2709   E.wrex 2710   _Vcvv 3017    i^i cin 3373    C_ wss 3374    |-> cmpt 4420   `'ccnv 4790   ran crn 4792   "cima 4794   Fun wfun 5533   -->wf 5535   ` cfv 5539  (class class class)co 6244   ficfi 7872   fBascfbas 18896   Filcfil 20797    FilMap cfm 20885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7520  df-fin 7523  df-fi 7873  df-fbas 18905  df-fil 20798
This theorem is referenced by:  fmfnfmlem4  20909
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