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Theorem fmfnfmlem3 17980
Description: Lemma for fmfnfm 17982. (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 17878 . . . . . . . . 9  |-  ( ( L  e.  ( Fil `  X )  /\  y  e.  L  /\  z  e.  L )  ->  (
y  i^i  z )  e.  L )
323expb 1154 . . . . . . . 8  |-  ( ( L  e.  ( Fil `  X )  /\  (
y  e.  L  /\  z  e.  L )
)  ->  ( y  i^i  z )  e.  L
)
41, 3sylan 458 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( y  i^i  z
)  e.  L )
5 fmfnfm.f . . . . . . . . . 10  |-  ( ph  ->  F : Y --> X )
6 ffun 5585 . . . . . . . . . 10  |-  ( F : Y --> X  ->  Fun  F )
75, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  Fun  F )
8 funcnvcnv 5501 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
9 imain 5521 . . . . . . . . . 10  |-  ( Fun  `' `' F  ->  ( `' F " ( y  i^i  z ) )  =  ( ( `' F " y )  i^i  ( `' F " z ) ) )
109eqcomd 2440 . . . . . . . . 9  |-  ( Fun  `' `' F  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
( y  i^i  z
) ) )
117, 8, 103syl 19 . . . . . . . 8  |-  ( ph  ->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
1211adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " ( y  i^i  z ) ) )
13 imaeq2 5191 . . . . . . . . 9  |-  ( x  =  ( y  i^i  z )  ->  ( `' F " x )  =  ( `' F " ( y  i^i  z
) ) )
1413eqeq2d 2446 . . . . . . . 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 ) ) ) )
1514rspcev 3044 . . . . . . 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 ) )
164, 12, 15syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  ->  E. x  e.  L  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F " x ) )
17 ineq12 3529 . . . . . . . 8  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( s  i^i  t )  =  ( ( `' F "
y )  i^i  ( `' F " z ) ) )
1817eqeq1d 2443 . . . . . . 7  |-  ( ( s  =  ( `' F " y )  /\  t  =  ( `' F " z ) )  ->  ( (
s  i^i  t )  =  ( `' F " x )  <->  ( ( `' F " y )  i^i  ( `' F " z ) )  =  ( `' F "
x ) ) )
1918rexbidv 2718 . . . . . 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 ) ) )
2016, 19syl5ibrcom 214 . . . . 5  |-  ( (
ph  /\  ( y  e.  L  /\  z  e.  L ) )  -> 
( ( s  =  ( `' F "
y )  /\  t  =  ( `' F " z ) )  ->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) ) )
2120rexlimdvva 2829 . . . 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 ) ) )
22 imaeq2 5191 . . . . . . . 8  |-  ( x  =  y  ->  ( `' F " x )  =  ( `' F " y ) )
2322eqeq2d 2446 . . . . . . 7  |-  ( x  =  y  ->  (
s  =  ( `' F " x )  <-> 
s  =  ( `' F " y ) ) )
2423cbvrexv 2925 . . . . . 6  |-  ( E. x  e.  L  s  =  ( `' F " x )  <->  E. y  e.  L  s  =  ( `' F " y ) )
25 imaeq2 5191 . . . . . . . 8  |-  ( x  =  z  ->  ( `' F " x )  =  ( `' F " z ) )
2625eqeq2d 2446 . . . . . . 7  |-  ( x  =  z  ->  (
t  =  ( `' F " x )  <-> 
t  =  ( `' F " z ) ) )
2726cbvrexv 2925 . . . . . 6  |-  ( E. x  e.  L  t  =  ( `' F " x )  <->  E. z  e.  L  t  =  ( `' F " z ) )
2824, 27anbi12i 679 . . . . 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 ) ) )
29 vex 2951 . . . . . . 7  |-  s  e. 
_V
30 eqid 2435 . . . . . . . 8  |-  ( x  e.  L  |->  ( `' F " x ) )  =  ( x  e.  L  |->  ( `' F " x ) )
3130elrnmpt 5109 . . . . . . 7  |-  ( s  e.  _V  ->  (
s  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  s  =  ( `' F " x ) ) )
3229, 31ax-mp 8 . . . . . 6  |-  ( s  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  s  =  ( `' F " x ) )
33 vex 2951 . . . . . . 7  |-  t  e. 
_V
3430elrnmpt 5109 . . . . . . 7  |-  ( t  e.  _V  ->  (
t  e.  ran  (
x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  t  =  ( `' F " x ) ) )
3533, 34ax-mp 8 . . . . . 6  |-  ( t  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  t  =  ( `' F " x ) )
3632, 35anbi12i 679 . . . . 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 ) ) )
37 reeanv 2867 . . . . 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 ) ) )
3828, 36, 373bitr4i 269 . . . 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 ) ) )
3929inex1 4336 . . . . 5  |-  ( s  i^i  t )  e. 
_V
4030elrnmpt 5109 . . . . 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 ) ) )
4139, 40ax-mp 8 . . . 4  |-  ( ( s  i^i  t )  e.  ran  ( x  e.  L  |->  ( `' F " x ) )  <->  E. x  e.  L  ( s  i^i  t
)  =  ( `' F " x ) )
4221, 38, 413imtr4g 262 . . 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 ) ) ) )
4342ralrimivv 2789 . 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 ) ) )
44 mptexg 5957 . . . 4  |-  ( L  e.  ( Fil `  X
)  ->  ( x  e.  L  |->  ( `' F " x ) )  e.  _V )
45 rnexg 5123 . . . 4  |-  ( ( x  e.  L  |->  ( `' F " x ) )  e.  _V  ->  ran  ( x  e.  L  |->  ( `' F "
x ) )  e. 
_V )
4644, 45syl 16 . . 3  |-  ( L  e.  ( Fil `  X
)  ->  ran  ( x  e.  L  |->  ( `' F " x ) )  e.  _V )
47 inficl 7422 . . 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 ) ) ) )
481, 46, 473syl 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 ) ) ) )
4943, 48mpbid 202 1  |-  ( ph  ->  ( fi `  ran  ( x  e.  L  |->  ( `' F "
x ) ) )  =  ran  ( x  e.  L  |->  ( `' F " x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312    e. cmpt 4258   `'ccnv 4869   ran crn 4871   "cima 4873   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   ficfi 7407   fBascfbas 16681   Filcfil 17869    FilMap cfm 17957
This theorem is referenced by:  fmfnfmlem4  17981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-fi 7408  df-fbas 16691  df-fil 17870
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