Theorem strfvssn 12640

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 12638	. 2 |- ( ph -> ( E ` S ) = ( S ` N ) )
5	3	elexd 2773	. . 3 |- ( ph -> N e. _V )
6		fvssunirng 5569	. . 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 3215	1 |- ( ph -> ( E ` S ) C_ U. ran S )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   U.cuni 3835   ran crn 4660   ` cfv 5254   NNcn 8982  Slot cslot 12617

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-slot 12622

This theorem is referenced by: (None)