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Theorem slotslfn 12001
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) ∈ ℕ)
Assertion
Ref Expression
slotslfn 𝐸 Fn V

Proof of Theorem slotslfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . 3 𝑥 ∈ V
2 slotslfn.e . . . 4 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
32simpri 112 . . 3 (𝐸‘ndx) ∈ ℕ
41, 3fvex 5441 . 2 (𝑥‘(𝐸‘ndx)) ∈ V
52simpli 110 . . 3 𝐸 = Slot (𝐸‘ndx)
6 df-slot 11979 . . 3 Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
75, 6eqtri 2160 . 2 𝐸 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
84, 7fnmpti 5251 1 𝐸 Fn V
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  cmpt 3989   Fn wfn 5118  cfv 5123  cn 8734  ndxcnx 11972  Slot cslot 11974
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-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-fn 5126  df-fv 5131  df-slot 11979
This theorem is referenced by:  slotex  12002  basfn  12032  topontopn  12220
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