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

Theorem elrfirn 26873
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 5411 . . 3  |-  ( F : I --> ~P B  ->  ran  F  C_  ~P B )
2 elrfi 26872 . . 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 460 . 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 5041 . . . . . 6  |-  ( F
" v )  C_  ran  F
5 pwexg 4210 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
6 ssexg 4176 . . . . . . . 8  |-  ( ( ran  F  C_  ~P B  /\  ~P B  e. 
_V )  ->  ran  F  e.  _V )
71, 5, 6syl2anr 464 . . . . . . 7  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ran  F  e. 
_V )
8 elpw2g 4190 . . . . . . 7  |-  ( ran 
F  e.  _V  ->  ( ( F " v
)  e.  ~P ran  F  <-> 
( F " v
)  C_  ran  F ) )
97, 8syl 15 . . . . . 6  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( ( F " v )  e. 
~P ran  F  <->  ( F " v )  C_  ran  F ) )
104, 9mpbiri 224 . . . . 5  |-  ( ( B  e.  V  /\  F : I --> ~P B
)  ->  ( F " v )  e.  ~P ran  F )
1110adantr 451 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( F " v )  e. 
~P ran  F )
12 ffun 5407 . . . . . 6  |-  ( F : I --> ~P B  ->  Fun  F )
1312ad2antlr 707 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  Fun  F )
14 inss2 3403 . . . . . . 7  |-  ( ~P I  i^i  Fin )  C_ 
Fin
1514sseli 3189 . . . . . 6  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  Fin )
1615adantl 452 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  v  e.  Fin )
17 imafi 7164 . . . . 5  |-  ( ( Fun  F  /\  v  e.  Fin )  ->  ( F " v )  e. 
Fin )
1813, 16, 17syl2anc 642 . . . 4  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  ( F " v )  e. 
Fin )
19 elin 3371 . . . 4  |-  ( ( F " v )  e.  ( ~P ran  F  i^i  Fin )  <->  ( ( F " v )  e. 
~P ran  F  /\  ( F " v )  e.  Fin ) )
2011, 18, 19sylanbrc 645 . . 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 5405 . . . . . 6  |-  ( F : I --> ~P B  ->  F  Fn  I )
2221ad2antlr 707 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  F  Fn  I )
23 inss1 3402 . . . . . . . 8  |-  ( ~P
ran  F  i^i  Fin )  C_ 
~P ran  F
2423sseli 3189 . . . . . . 7  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  ~P ran  F )
25 elpwi 3646 . . . . . . 7  |-  ( w  e.  ~P ran  F  ->  w  C_  ran  F )
2624, 25syl 15 . . . . . 6  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  C_  ran  F )
2726adantl 452 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  w  C_ 
ran  F )
28 inss2 3403 . . . . . . 7  |-  ( ~P
ran  F  i^i  Fin )  C_ 
Fin
2928sseli 3189 . . . . . 6  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  Fin )
3029adantl 452 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  e.  ( ~P ran  F  i^i  Fin ) )  ->  w  e.  Fin )
31 fipreima 7177 . . . . 5  |-  ( ( F  Fn  I  /\  w  C_  ran  F  /\  w  e.  Fin )  ->  E. v  e.  ( ~P I  i^i  Fin ) ( F "
v )  =  w )
3222, 27, 30, 31syl3anc 1182 . . . 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 )
33 eqcom 2298 . . . . 5  |-  ( ( F " v )  =  w  <->  w  =  ( F " v ) )
3433rexbii 2581 . . . 4  |-  ( E. v  e.  ( ~P I  i^i  Fin )
( F " v
)  =  w  <->  E. v  e.  ( ~P I  i^i 
Fin ) w  =  ( F " v
) )
3532, 34sylib 188 . . 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
) )
36 inteq 3881 . . . . . 6  |-  ( w  =  ( F "
v )  ->  |^| w  =  |^| ( F "
v ) )
3736ineq2d 3383 . . . . 5  |-  ( w  =  ( F "
v )  ->  ( B  i^i  |^| w )  =  ( B  i^i  |^| ( F " v ) ) )
3837eqeq2d 2307 . . . 4  |-  ( w  =  ( F "
v )  ->  ( A  =  ( B  i^i  |^| w )  <->  A  =  ( B  i^i  |^| ( F " v ) ) ) )
3938adantl 452 . . 3  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  w  =  ( F " v ) )  ->  ( A  =  ( B  i^i  |^| w )  <->  A  =  ( B  i^i  |^| ( F " v ) ) ) )
4020, 35, 39rexxfrd 4565 . 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 ) ) ) )
4121ad2antlr 707 . . . . . . 7  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  F  Fn  I )
42 inss1 3402 . . . . . . . . . 10  |-  ( ~P I  i^i  Fin )  C_ 
~P I
4342sseli 3189 . . . . . . . . 9  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  ~P I )
44 elpwi 3646 . . . . . . . . 9  |-  ( v  e.  ~P I  -> 
v  C_  I )
4543, 44syl 15 . . . . . . . 8  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  C_  I )
4645adantl 452 . . . . . . 7  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  v  C_  I )
47 imaiinfv 26862 . . . . . . 7  |-  ( ( F  Fn  I  /\  v  C_  I )  ->  |^|_ y  e.  v  ( F `  y )  =  |^| ( F
" v ) )
4841, 46, 47syl2anc 642 . . . . . 6  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^|_ y  e.  v  ( F `  y )  =  |^| ( F " v ) )
4948eqcomd 2301 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^| ( F " v )  = 
|^|_ y  e.  v  ( F `  y
) )
5049ineq2d 3383 . . . 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 )
) )
5150eqeq2d 2307 . . 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 ) ) ) )
5251rexbidva 2573 . 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 )
) ) )
533, 40, 523bitrd 270 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   |^|cint 3878   |^|_ciin 3922   ran crn 4706   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271   Fincfn 6879   ficfi 7180
This theorem is referenced by:  elrfirn2  26874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883  df-fi 7181
  Copyright terms: Public domain W3C validator