Theorem strfvssn 11970

Description: A structure component extractor produces a value which is contained in a set dependent on S, but not E. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)

Hypotheses

Ref	Expression
strfvssn.c	|- E = Slot N
strfvssn.s	|- ( ph -> S e. V )
strfvssn.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
strfvssn	|- ( ph -> ( E ` S ) C_ U. ran S )

Proof of Theorem strfvssn

Step	Hyp	Ref	Expression
1		strfvssn.c	. . 3 |- E = Slot N
2		strfvssn.s	. . 3 |- ( ph -> S e. V )
3		strfvssn.n	. . 3 |- ( ph -> N e. NN )
4	1, 2, 3	strnfvnd 11968	. 2 |- ( ph -> ( E ` S ) = ( S ` N ) )
5	3	elexd 2694	. . 3 |- ( ph -> N e. _V )
6		fvssunirng 5429	. . 3 |- ( N e. _V -> ( S ` N ) C_ U. ran S )
7	5, 6	syl 14	. 2 |- ( ph -> ( S ` N ) C_ U. ran S )
8	4, 7	eqsstrd 3128	1 |- ( ph -> ( E ` S ) C_ U. ran S )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2681   C_ wss 3066   U.cuni 3731   ran crn 4535   ` cfv 5118   NNcn 8713  Slot cslot 11947

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-slot 11952

This theorem is referenced by: (None)