Theorem strslfv2 11686

Description: A variation on strslfv 11687 to avoid asserting that S itself is a function, which involves sethood of all the ordered pair components of S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv2.s	|- S e. _V
strfv2.f	|- Fun `' `' S
strslfv2.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfv2.n	|- <. ( E ` ndx ) , C >. e. S

Assertion

Ref	Expression
strslfv2	|- ( C e. V -> C = ( E ` S ) )

Proof of Theorem strslfv2

Step	Hyp	Ref	Expression
1		strslfv2.e	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strfv2.s	. . 3 |- S e. _V
3	2	a1i 9	. 2 |- ( C e. V -> S e. _V )
4		strfv2.f	. . 3 |- Fun `' `' S
5	4	a1i 9	. 2 |- ( C e. V -> Fun `' `' S )
6		strfv2.n	. . 3 |- <. ( E ` ndx ) , C >. e. S
7	6	a1i 9	. 2 |- ( C e. V -> <. ( E ` ndx ) , C >. e. S )
8		id 19	. 2 |- ( C e. V -> C e. V )
9	1, 3, 5, 7, 8	strslfv2d 11685	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   _Vcvv 2633   <.cop 3469   `'ccnv 4466   Fun wfun 5043   ` cfv 5049   NNcn 8520   ndxcnx 11640  Slot cslot 11642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fv 5057  df-slot 11647

This theorem is referenced by: (None)