ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  slotslfn GIF version

Theorem slotslfn 12458
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 2740 . . 3 𝑥 ∈ V
2 slotslfn.e . . . 4 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
32simpri 113 . . 3 (𝐸‘ndx) ∈ ℕ
41, 3fvex 5530 . 2 (𝑥‘(𝐸‘ndx)) ∈ V
52simpli 111 . . 3 𝐸 = Slot (𝐸‘ndx)
6 df-slot 12436 . . 3 Slot (𝐸‘ndx) = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
75, 6eqtri 2198 . 2 𝐸 = (𝑥 ∈ V ↦ (𝑥‘(𝐸‘ndx)))
84, 7fnmpti 5339 1 𝐸 Fn V
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cmpt 4061   Fn wfn 5206  cfv 5211  cn 8895  ndxcnx 12429  Slot cslot 12431
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-slot 12436
This theorem is referenced by:  slotex  12459  basfn  12489  topontopn  13168
  Copyright terms: Public domain W3C validator