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Theorem slotslfn 12022
Description: A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
Hypothesis
Ref Expression
slotslfn.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
Assertion
Ref Expression
slotslfn  |-  E  Fn  _V

Proof of Theorem slotslfn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2692 . . 3  |-  x  e. 
_V
2 slotslfn.e . . . 4  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
32simpri 112 . . 3  |-  ( E `
 ndx )  e.  NN
41, 3fvex 5448 . 2  |-  ( x `
 ( E `  ndx ) )  e.  _V
52simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
6 df-slot 12000 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
75, 6eqtri 2161 . 2  |-  E  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
84, 7fnmpti 5258 1  |-  E  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2689    |-> cmpt 3996    Fn wfn 5125   ` cfv 5130   NNcn 8743   ndxcnx 11993  Slot cslot 11995
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 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
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-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-slot 12000
This theorem is referenced by:  slotex  12023  basfn  12053  topontopn  12241
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