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Theorem slotslfn 11912
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 2663 . . 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 5409 . 2  |-  ( x `
 ( E `  ndx ) )  e.  _V
52simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
6 df-slot 11890 . . 3  |- Slot  ( E `
 ndx )  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
75, 6eqtri 2138 . 2  |-  E  =  ( x  e.  _V  |->  ( x `  ( E `  ndx ) ) )
84, 7fnmpti 5221 1  |-  E  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660    |-> cmpt 3959    Fn wfn 5088   ` cfv 5093   NNcn 8688   ndxcnx 11883  Slot cslot 11885
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-slot 11890
This theorem is referenced by:  slotex  11913  basfn  11943  topontopn  12131
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