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Theorem strfvssn 12464
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 12462 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 N ) )
53elexd 2750 . . 3  |-  ( ph  ->  N  e.  _V )
6 fvssunirng 5526 . . 3  |-  ( N  e.  _V  ->  ( S `  N )  C_ 
U. ran  S )
75, 6syl 14 . 2  |-  ( ph  ->  ( S `  N
)  C_  U. ran  S
)
84, 7eqsstrd 3191 1  |-  ( ph  ->  ( E `  S
)  C_  U. ran  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   U.cuni 3807   ran crn 4624   ` cfv 5212   NNcn 8905  Slot cslot 12441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fv 5220  df-slot 12446
This theorem is referenced by: (None)
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