Theorem strfvssn 12825

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 12823	. 2 |- ( ph -> ( E ` S ) = ( S ` N ) )
5	3	elexd 2784	. . 3 |- ( ph -> N e. _V )
6		fvssunirng 5590	. . 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 3228	1 |- ( ph -> ( E ` S ) C_ U. ran S )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   U.cuni 3849   ran crn 4675   ` cfv 5270   NNcn 9035  Slot cslot 12802

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fv 5278  df-slot 12807

This theorem is referenced by:  prdsvallem  13075  prdsval  13076