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Theorem strfvss 13479
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 20 . . . 4  |-  ( S  e.  _V  ->  S  e.  _V )
31, 2strfvnd 13476 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
4 fvssunirn 5746 . . 3  |-  ( S `
 N )  C_  U.
ran  S
53, 4syl6eqss 3390 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
6 fvprc 5714 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
7 0ss 3648 . . 3  |-  (/)  C_  U. ran  S
86, 7syl6eqss 3390 . 2  |-  ( -.  S  e.  _V  ->  ( E `  S ) 
C_  U. ran  S )
95, 8pm2.61i 158 1  |-  ( E `
 S )  C_  U.
ran  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   (/)c0 3620   U.cuni 4007   ran crn 4871   ` cfv 5446  Slot cslot 13460
This theorem is referenced by:  wunstr  13480  prdsval  13670  prdsbas  13672  prdsplusg  13673  prdsmulr  13674  prdsvsca  13675  prdshom  13681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-slot 13465
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