Theorem strslfv2 13189

Description: A variation on strslfv 13190 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 13188	1 |- ( C e. V -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676   `'ccnv 4730   Fun wfun 5327   ` cfv 5333   NNcn 9185   ndxcnx 13142  Slot cslot 13144

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-slot 13149

This theorem is referenced by: (None)