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Theorem elrfirn 26770
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 5395 . . 3  |-  ( F : I --> ~P B  ->  ran  F  C_  ~P B )
2 elrfi 26769 . . 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 5025 . . . . . 6  |-  ( F
" v )  C_  ran  F
5 pwexg 4194 . . . . . . . 8  |-  ( B  e.  V  ->  ~P B  e.  _V )
6 ssexg 4160 . . . . . . . 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 4174 . . . . . . 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 5391 . . . . . 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 3390 . . . . . . 7  |-  ( ~P I  i^i  Fin )  C_ 
Fin
1514sseli 3176 . . . . . 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 7148 . . . . 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 3358 . . . 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 5389 . . . . . 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 3389 . . . . . . . 8  |-  ( ~P
ran  F  i^i  Fin )  C_ 
~P ran  F
2423sseli 3176 . . . . . . 7  |-  ( w  e.  ( ~P ran  F  i^i  Fin )  ->  w  e.  ~P ran  F )
25 elpwi 3633 . . . . . . 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 3390 . . . . . . 7  |-  ( ~P
ran  F  i^i  Fin )  C_ 
Fin
2928sseli 3176 . . . . . 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 7161 . . . . 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 2285 . . . . 5  |-  ( ( F " v )  =  w  <->  w  =  ( F " v ) )
3433rexbii 2568 . . . 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 3865 . . . . . 6  |-  ( w  =  ( F "
v )  ->  |^| w  =  |^| ( F "
v ) )
3736ineq2d 3370 . . . . 5  |-  ( w  =  ( F "
v )  ->  ( B  i^i  |^| w )  =  ( B  i^i  |^| ( F " v ) ) )
3837eqeq2d 2294 . . . 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 4549 . 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 3389 . . . . . . . . . 10  |-  ( ~P I  i^i  Fin )  C_ 
~P I
4342sseli 3176 . . . . . . . . 9  |-  ( v  e.  ( ~P I  i^i  Fin )  ->  v  e.  ~P I )
44 elpwi 3633 . . . . . . . . 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 26759 . . . . . . 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 2288 . . . . 5  |-  ( ( ( B  e.  V  /\  F : I --> ~P B
)  /\  v  e.  ( ~P I  i^i  Fin ) )  ->  |^| ( F " v )  = 
|^|_ y  e.  v  ( F `  y
) )
5049ineq2d 3370 . . . 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 2294 . . 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 2560 . 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 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   |^|cint 3862   |^|_ciin 3906   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   Fincfn 6863   ficfi 7164
This theorem is referenced by:  elrfirn2  26771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867  df-fi 7165
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