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Theorem strslfv2 12519
Description: A variation on strslfv 12520 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 12518 1  |-  ( C  e.  V  ->  C  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   _Vcvv 2749   <.cop 3607   `'ccnv 4637   Fun wfun 5222   ` cfv 5228   NNcn 8932   ndxcnx 12472  Slot cslot 12474
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fv 5236  df-slot 12479
This theorem is referenced by: (None)
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