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Theorem elrfirn 26639
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 5556 . . 3  |-  ( F : I --> ~P B  ->  ran  F  C_  ~P B )
2 elrfi 26638 . . 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 5175 . . . . . 6  |-  ( F
" v )  C_  ran  F
5 pwexg 4343 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
6 ssexg 4309 . . . . . . . 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 4323 . . . . . . 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 5552 . . . . . 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 3522 . . . . . . 7  |-  ( ~P I  i^i  Fin )  C_ 
Fin
1514sseli 3304 . . . . . 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 7357 . . . . 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 3490 . . . 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 5550 . . . . . 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 3521 . . . . . . . 8  |-  ( ~P
ran  F  i^i  Fin )  C_ 
~P ran  F
2423sseli 3304 . . . . . . 7  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  ~P ran  F )
2524elpwid 3768 . . . . . 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 3522 . . . . . . 7  |-  ( ~P
ran  F  i^i  Fin )  C_ 
Fin
2827sseli 3304 . . . . . 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 7370 . . . . 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 2406 . . . . 5  |-  ( ( F " v )  =  w  <->  w  =  ( F " v ) )
3332rexbii 2691 . . . 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 4013 . . . . . 6  |-  ( w  =  ( F "
v )  ->  |^| w  =  |^| ( F "
v ) )
3635ineq2d 3502 . . . . 5  |-  ( w  =  ( F "
v )  ->  ( B  i^i  |^| w )  =  ( B  i^i  |^| ( F " v ) ) )
3736eqeq2d 2415 . . . 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 4697 . 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 3521 . . . . . . . . . 10  |-  ( ~P I  i^i  Fin )  C_ 
~P I
4241sseli 3304 . . . . . . . . 9  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  ~P I )
4342elpwid 3768 . . . . . . . 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 26630 . . . . . . 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 2409 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^| ( F " v )  = 
|^|_ y  e.  v  ( F `  y
) )
4847ineq2d 3502 . . . 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 2415 . . 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 2683 . 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 1721   E.wrex 2667   _Vcvv 2916    u. cun 3278    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   {csn 3774   |^|cint 4010   |^|_ciin 4054   ran crn 4838   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413   Fincfn 7068   ficfi 7373
This theorem is referenced by:  elrfirn2  26640
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-fin 7072  df-fi 7374
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