Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrfirn Unicode version

Theorem elrfirn 26442
Description: Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
elrfirn  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. v  e.  ( ~P I  i^i 
Fin ) A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y )
) ) )
Distinct variable groups:    v, A    v, B    v, F, y   
v, I    v, V    y, v
Allowed substitution hints:    A( y)    B( y)    I( y)    V( y)

Proof of Theorem elrfirn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 frn 5539 . . 3  |-  ( F : I --> ~P B  ->  ran  F  C_  ~P B )
2 elrfi 26441 . . 3  |-  ( ( B  e.  V  /\  ran  F  C_  ~P B
)  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. w  e.  ( ~P ran  F  i^i  Fin ) A  =  ( B  i^i  |^| w ) ) )
31, 2sylan2 461 . 2  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. w  e.  ( ~P ran  F  i^i  Fin ) A  =  ( B  i^i  |^| w ) ) )
4 imassrn 5158 . . . . . 6  |-  ( F
" v )  C_  ran  F
5 pwexg 4326 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
6 ssexg 4292 . . . . . . . 8  |-  ( ( ran  F  C_  ~P B  /\  ~P B  e. 
_V )  ->  ran  F  e.  _V )
71, 5, 6syl2anr 465 . . . . . . 7  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ran  F  e. 
_V )
8 elpw2g 4306 . . . . . . 7  |-  ( ran 
F  e.  _V  ->  ( ( F " v
)  e.  ~P ran  F  <-> 
( F " v
)  C_  ran  F ) )
97, 8syl 16 . . . . . 6  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( ( F " v )  e. 
~P ran  F  <->  ( F " v )  C_  ran  F ) )
104, 9mpbiri 225 . . . . 5  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( F " v )  e.  ~P ran  F )
1110adantr 452 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( F " v )  e. 
~P ran  F )
12 ffun 5535 . . . . . 6  |-  ( F : I --> ~P B  ->  Fun  F )
1312ad2antlr 708 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  Fun  F )
14 inss2 3507 . . . . . . 7  |-  ( ~P I  i^i  Fin )  C_ 
Fin
1514sseli 3289 . . . . . 6  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  Fin )
1615adantl 453 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  v  e.  Fin )
17 imafi 7336 . . . . 5  |-  ( ( Fun  F  /\  v  e.  Fin )  ->  ( F " v )  e. 
Fin )
1813, 16, 17syl2anc 643 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( F " v )  e. 
Fin )
19 elin 3475 . . . 4  |-  ( ( F " v )  e.  ( ~P ran  F  i^i  Fin )  <->  ( ( F " v )  e. 
~P ran  F  /\  ( F " v )  e.  Fin ) )
2011, 18, 19sylanbrc 646 . . 3  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( F " v )  e.  ( ~P ran  F  i^i  Fin ) )
21 ffn 5533 . . . . . 6  |-  ( F : I --> ~P B  ->  F  Fn  I )
2221ad2antlr 708 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  F  Fn  I )
23 inss1 3506 . . . . . . . 8  |-  ( ~P
ran  F  i^i  Fin )  C_ 
~P ran  F
2423sseli 3289 . . . . . . 7  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  ~P ran  F )
2524elpwid 3753 . . . . . 6  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  C_  ran  F )
2625adantl 453 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  w  C_ 
ran  F )
27 inss2 3507 . . . . . . 7  |-  ( ~P
ran  F  i^i  Fin )  C_ 
Fin
2827sseli 3289 . . . . . 6  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  Fin )
2928adantl 453 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  w  e.  Fin )
30 fipreima 7349 . . . . 5  |-  ( ( F  Fn  I  /\  w  C_  ran  F  /\  w  e.  Fin )  ->  E. v  e.  ( ~P I  i^i  Fin ) ( F "
v )  =  w )
3122, 26, 29, 30syl3anc 1184 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  E. v  e.  ( ~P I  i^i 
Fin ) ( F
" v )  =  w )
32 eqcom 2391 . . . . 5  |-  ( ( F " v )  =  w  <->  w  =  ( F " v ) )
3332rexbii 2676 . . . 4  |-  ( E. v  e.  ( ~P I  i^i  Fin )
( F " v
)  =  w  <->  E. v  e.  ( ~P I  i^i 
Fin ) w  =  ( F " v
) )
3431, 33sylib 189 . . 3  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  E. v  e.  ( ~P I  i^i 
Fin ) w  =  ( F " v
) )
35 inteq 3997 . . . . . 6  |-  ( w  =  ( F "
v )  ->  |^| w  =  |^| ( F "
v ) )
3635ineq2d 3487 . . . . 5  |-  ( w  =  ( F "
v )  ->  ( B  i^i  |^| w )  =  ( B  i^i  |^| ( F " v ) ) )
3736eqeq2d 2400 . . . 4  |-  ( w  =  ( F "
v )  ->  ( A  =  ( B  i^i  |^| w )  <->  A  =  ( B  i^i  |^| ( F " v ) ) ) )
3837adantl 453 . . 3  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  =  ( F " v ) )  ->  ( A  =  ( B  i^i  |^| w )  <->  A  =  ( B  i^i  |^| ( F " v ) ) ) )
3920, 34, 38rexxfrd 4680 . 2  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( E. w  e.  ( ~P ran  F  i^i  Fin ) A  =  ( B  i^i  |^| w )  <->  E. v  e.  ( ~P I  i^i 
Fin ) A  =  ( B  i^i  |^| ( F " v ) ) ) )
4021ad2antlr 708 . . . . . . 7  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  F  Fn  I )
41 inss1 3506 . . . . . . . . . 10  |-  ( ~P I  i^i  Fin )  C_ 
~P I
4241sseli 3289 . . . . . . . . 9  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  ~P I )
4342elpwid 3753 . . . . . . . 8  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  C_  I )
4443adantl 453 . . . . . . 7  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  v  C_  I )
45 imaiinfv 26433 . . . . . . 7  |-  ( ( F  Fn  I  /\  v  C_  I )  ->  |^|_ y  e.  v  ( F `  y )  =  |^| ( F
" v ) )
4640, 44, 45syl2anc 643 . . . . . 6  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^|_ y  e.  v  ( F `  y )  =  |^| ( F " v ) )
4746eqcomd 2394 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^| ( F " v )  = 
|^|_ y  e.  v  ( F `  y
) )
4847ineq2d 3487 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( B  i^i  |^| ( F "
v ) )  =  ( B  i^i  |^|_ y  e.  v  ( F `  y )
) )
4948eqeq2d 2400 . . 3  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( A  =  ( B  i^i  |^| ( F "
v ) )  <->  A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y ) ) ) )
5049rexbidva 2668 . 2  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( E. v  e.  ( ~P I  i^i  Fin ) A  =  ( B  i^i  |^| ( F " v
) )  <->  E. v  e.  ( ~P I  i^i 
Fin ) A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y )
) ) )
513, 39, 503bitrd 271 1  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. v  e.  ( ~P I  i^i 
Fin ) A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652   _Vcvv 2901    u. cun 3263    i^i cin 3264    C_ wss 3265   ~Pcpw 3744   {csn 3759   |^|cint 3994   |^|_ciin 4038   ran crn 4821   "cima 4823   Fun wfun 5390    Fn wfn 5391   -->wf 5392   ` cfv 5396   Fincfn 7047   ficfi 7352
This theorem is referenced by:  elrfirn2  26443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-fin 7051  df-fi 7353
  Copyright terms: Public domain W3C validator