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Theorem strfvssn 12011
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.) (Revised by Jim Kingdon, 19-Jan-2023.)
Hypotheses
Ref Expression
strfvssn.c  |-  E  = Slot 
N
strfvssn.s  |-  ( ph  ->  S  e.  V )
strfvssn.n  |-  ( ph  ->  N  e.  NN )
Assertion
Ref Expression
strfvssn  |-  ( ph  ->  ( E `  S
)  C_  U. ran  S
)

Proof of Theorem strfvssn
StepHypRef Expression
1 strfvssn.c . . 3  |-  E  = Slot 
N
2 strfvssn.s . . 3  |-  ( ph  ->  S  e.  V )
3 strfvssn.n . . 3  |-  ( ph  ->  N  e.  NN )
41, 2, 3strnfvnd 12009 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
53elexd 2700 . . 3  |-  ( ph  ->  N  e.  _V )
6 fvssunirng 5440 . . 3  |-  ( N  e.  _V  ->  ( S `  N )  C_ 
U. ran  S )
75, 6syl 14 . 2  |-  ( ph  ->  ( S `  N
)  C_  U. ran  S
)
84, 7eqsstrd 3134 1  |-  ( ph  ->  ( E `  S
)  C_  U. ran  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   _Vcvv 2687    C_ wss 3072   U.cuni 3740   ran crn 4544   ` cfv 5127   NNcn 8740  Slot cslot 11988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-iota 5092  df-fun 5129  df-fv 5135  df-slot 11993
This theorem is referenced by: (None)
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