Theorem slotslfn 12357

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 2724	. . 3 |- x e. _V
2		slotslfn.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3	2	simpri 112	. . 3 |- ( E ` ndx ) e. NN
4	1, 3	fvex 5500	. 2 |- ( x ` ( E ` ndx ) ) e. _V
5	2	simpli 110	. . 3 |- E = Slot ( E ` ndx )
6		df-slot 12335	. . 3 |- Slot ( E ` ndx ) = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
7	5, 6	eqtri 2185	. 2 |- E = ( x e. _V |-> ( x ` ( E ` ndx ) ) )
8	4, 7	fnmpti 5310	1 |- E Fn _V

Syntax hints:   /\ wa 103   = wceq 1342   e. wcel 2135   _Vcvv 2721   |-> cmpt 4037   Fn wfn 5177   ` cfv 5182   NNcn 8848   ndxcnx 12328  Slot cslot 12330

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-fv 5190  df-slot 12335

This theorem is referenced by:  slotex  12358  basfn  12388  topontopn  12576