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Theorem strfvss 14508
Description: A structure component extractor produces a value which is contained in a set dependent on  S, but not  E. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
ndxarg.1  |-  E  = Slot 
N
Assertion
Ref Expression
strfvss  |-  ( E `
 S )  C_  U.
ran  S

Proof of Theorem strfvss
StepHypRef Expression
1 ndxarg.1 . . . 4  |-  E  = Slot 
N
2 id 22 . . . 4  |-  ( S  e.  _V  ->  S  e.  _V )
31, 2strfvnd 14505 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
4 fvssunirn 5889 . . 3  |-  ( S `
 N )  C_  U.
ran  S
53, 4syl6eqss 3554 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
6 fvprc 5860 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
7 0ss 3814 . . 3  |-  (/)  C_  U. ran  S
86, 7syl6eqss 3554 . 2  |-  ( -.  S  e.  _V  ->  ( E `  S ) 
C_  U. ran  S )
95, 8pm2.61i 164 1  |-  ( E `
 S )  C_  U.
ran  S
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   U.cuni 4245   ran crn 5000   ` cfv 5588  Slot cslot 14489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-slot 14494
This theorem is referenced by:  wunstr  14509  prdsval  14710  prdsbas  14712  prdsplusg  14713  prdsmulr  14714  prdsvsca  14715  prdshom  14722
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