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Theorem strslfv2 12041
Description: A variation on strslfv 12042 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 12040 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2689   <.cop 3535   `'ccnv 4546   Fun wfun 5125   ` cfv 5131   NNcn 8744   ndxcnx 11995  Slot cslot 11997
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 4054  ax-pow 4106  ax-pr 4139  ax-un 4363
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 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fv 5139  df-slot 12002
This theorem is referenced by: (None)
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