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Theorem strslfv2 12524
Description: A variation on strslfv 12525 to avoid asserting that  S itself is a function, which involves sethood of all the ordered pair components of  S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strfv2.s  |-  S  e. 
_V
strfv2.f  |-  Fun  `' `' S
strslfv2.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
strfv2.n  |-  <. ( E `  ndx ) ,  C >.  e.  S
Assertion
Ref Expression
strslfv2  |-  ( C  e.  V  ->  C  =  ( E `  S ) )

Proof of Theorem strslfv2
StepHypRef Expression
1 strslfv2.e . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
2 strfv2.s . . 3  |-  S  e. 
_V
32a1i 9 . 2  |-  ( C  e.  V  ->  S  e.  _V )
4 strfv2.f . . 3  |-  Fun  `' `' S
54a1i 9 . 2  |-  ( C  e.  V  ->  Fun  `' `' S )
6 strfv2.n . . 3  |-  <. ( E `  ndx ) ,  C >.  e.  S
76a1i 9 . 2  |-  ( C  e.  V  ->  <. ( E `  ndx ) ,  C >.  e.  S
)
8 id 19 . 2  |-  ( C  e.  V  ->  C  e.  V )
91, 3, 5, 7, 8strslfv2d 12523 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   <.cop 3610   `'ccnv 4640   Fun wfun 5225   ` cfv 5231   NNcn 8937   ndxcnx 12477  Slot cslot 12479
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-slot 12484
This theorem is referenced by: (None)
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