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Theorem strfvss 13182
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 fvssunirn 5567 . . 3  |-  ( S `
 N )  C_  U.
ran  S
2 ndxarg.1 . . . . 5  |-  E  = Slot 
N
3 id 19 . . . . 5  |-  ( S  e.  _V  ->  S  e.  _V )
42, 3strfvnd 13179 . . . 4  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
54sseq1d 3218 . . 3  |-  ( S  e.  _V  ->  (
( E `  S
)  C_  U. ran  S  <->  ( S `  N ) 
C_  U. ran  S ) )
61, 5mpbiri 224 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
7 0ss 3496 . . 3  |-  (/)  C_  U. ran  S
8 fvprc 5535 . . . 4  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
98sseq1d 3218 . . 3  |-  ( -.  S  e.  _V  ->  ( ( E `  S
)  C_  U. ran  S  <->  (/)  C_ 
U. ran  S )
)
107, 9mpbiri 224 . 2  |-  ( -.  S  e.  _V  ->  ( E `  S ) 
C_  U. ran  S )
116, 10pm2.61i 156 1  |-  ( E `
 S )  C_  U.
ran  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U.cuni 3843   ran crn 4706   ` cfv 5271  Slot cslot 13163
This theorem is referenced by:  wunstr  13183  prdsval  13371  prdsbas  13373  prdsplusg  13374  prdsmulr  13375  prdsvsca  13376  prdshom  13382
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168
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