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Theorem strslfvd 12014
Description: Deduction version of strslfv 12017. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strslfvd.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
strfvd.s  |-  ( ph  ->  S  e.  V )
strfvd.f  |-  ( ph  ->  Fun  S )
strfvd.n  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
Assertion
Ref Expression
strslfvd  |-  ( ph  ->  C  =  ( E `
 S ) )

Proof of Theorem strslfvd
StepHypRef Expression
1 strslfvd.e . . . 4  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
21simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
3 strfvd.s . . 3  |-  ( ph  ->  S  e.  V )
41simpri 112 . . . 4  |-  ( E `
 ndx )  e.  NN
54a1i 9 . . 3  |-  ( ph  ->  ( E `  ndx )  e.  NN )
62, 3, 5strnfvnd 11993 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
7 strfvd.f . . 3  |-  ( ph  ->  Fun  S )
8 strfvd.n . . 3  |-  ( ph  -> 
<. ( E `  ndx ) ,  C >.  e.  S )
9 funopfv 5461 . . 3  |-  ( Fun 
S  ->  ( <. ( E `  ndx ) ,  C >.  e.  S  ->  ( S `  ( E `  ndx ) )  =  C ) )
107, 8, 9sylc 62 . 2  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  C )
116, 10eqtr2d 2173 1  |-  ( ph  ->  C  =  ( E `
 S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   <.cop 3530   Fun wfun 5117   ` cfv 5123   NNcn 8732   ndxcnx 11970  Slot cslot 11972
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-slot 11977
This theorem is referenced by:  strslssd  12019  1strbas  12072  2strbasg  12074  2stropg  12075
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