Theorem slotslfn 13359

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.

Step	Hyp	Ref	Expression
1		vex 2824	. . 3 |- x e. _V
2		slotslfn.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpri 113	. . 3 |- ( E ` ndx ) e. NN
4	1, 3	fvex 5713	. 2 |- ( x ` ( E ` ndx ) ) e. _V
5	2	simpli 111	. . 3 |- E = Slot ( E ` ndx )
6		df-slot 13337	. . 3 |- Slot ( E ` ndx ) = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
7	5, 6	eqtri 2259	. 2 |- E = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
8	4, 7	fnmpti 5510	1 |- E Fn _V

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4190   Fn wfn 5370   ` cfv 5375   NNcn 9286   ndxcnx 13330  Slot cslot 13332

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-slot 13337

This theorem is referenced by:  slotex  13360  basfn  13392  topontopn  15064