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Theorem strfvss 14191
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 14188 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
4 fvssunirn 5712 . . 3  |-  ( S `
 N )  C_  U.
ran  S
53, 4syl6eqss 3405 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
6 fvprc 5684 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
7 0ss 3665 . . 3  |-  (/)  C_  U. ran  S
86, 7syl6eqss 3405 . 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 1369    e. wcel 1756   _Vcvv 2971    C_ wss 3327   (/)c0 3636   U.cuni 4090   ran crn 4840   ` cfv 5417  Slot cslot 14172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-slot 14177
This theorem is referenced by:  wunstr  14192  prdsval  14392  prdsbas  14394  prdsplusg  14395  prdsmulr  14396  prdsvsca  14397  prdshom  14404
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