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Theorem strfvss 15102
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 23 . . . 4  |-  ( S  e.  _V  ->  S  e.  _V )
31, 2strfvnd 15099 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
4 fvssunirn 5904 . . 3  |-  ( S `
 N )  C_  U.
ran  S
53, 4syl6eqss 3520 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
6 fvprc 5875 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
7 0ss 3797 . . 3  |-  (/)  C_  U. ran  S
86, 7syl6eqss 3520 . 2  |-  ( -.  S  e.  _V  ->  ( E `  S ) 
C_  U. ran  S )
95, 8pm2.61i 167 1  |-  ( E `
 S )  C_  U.
ran  S
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   (/)c0 3767   U.cuni 4222   ran crn 4855   ` cfv 5601  Slot cslot 15083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-slot 15088
This theorem is referenced by:  wunstr  15103  prdsval  15312  prdsbas  15314  prdsplusg  15315  prdsmulr  15316  prdsvsca  15317  prdshom  15324
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