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Theorem strfvss 13492
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 21 . . . 4  |-  ( S  e.  _V  ->  S  e.  _V )
31, 2strfvnd 13489 . . 3  |-  ( S  e.  _V  ->  ( E `  S )  =  ( S `  N ) )
4 fvssunirn 5757 . . 3  |-  ( S `
 N )  C_  U.
ran  S
53, 4syl6eqss 3400 . 2  |-  ( S  e.  _V  ->  ( E `  S )  C_ 
U. ran  S )
6 fvprc 5725 . . 3  |-  ( -.  S  e.  _V  ->  ( E `  S )  =  (/) )
7 0ss 3658 . . 3  |-  (/)  C_  U. ran  S
86, 7syl6eqss 3400 . 2  |-  ( -.  S  e.  _V  ->  ( E `  S ) 
C_  U. ran  S )
95, 8pm2.61i 159 1  |-  ( E `
 S )  C_  U.
ran  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   (/)c0 3630   U.cuni 4017   ran crn 4882   ` cfv 5457  Slot cslot 13473
This theorem is referenced by:  wunstr  13493  prdsval  13683  prdsbas  13685  prdsplusg  13686  prdsmulr  13687  prdsvsca  13688  prdshom  13694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478
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