Theorem igenidl 33862

Description: The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
igenval.1	|- G = ( 1st ` R )
igenval.2	|- X = ran G

Assertion

Ref	Expression
igenidl	|- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) e. ( Idl ` R ) )

Proof of Theorem igenidl

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval.1	. . 3 |- G = ( 1st ` R )
2		igenval.2	. . 3 |- X = ran G
3	1, 2	igenval 33860	. 2 |- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) = |^| { j e. ( Idl ` R ) | S C_ j } )
4	1, 2	rngoidl 33823	. . . . 5 |- ( R e. RingOps -> X e. ( Idl ` R ) )
5		sseq2 3627	. . . . . 6 |- ( j = X -> ( S C_ j <-> S C_ X ) )
6	5	rspcev 3309	. . . . 5 |- ( ( X e. ( Idl ` R ) /\ S C_ X ) -> E. j e. ( Idl ` R ) S C_ j )
7	4, 6	sylan 488	. . . 4 |- ( ( R e. RingOps /\ S C_ X ) -> E. j e. ( Idl ` R ) S C_ j )
8		rabn0 3958	. . . 4 |- ( { j e. ( Idl ` R ) | S C_ j } =/= (/) <-> E. j e. ( Idl ` R ) S C_ j )
9	7, 8	sylibr 224	. . 3 |- ( ( R e. RingOps /\ S C_ X ) -> { j e. ( Idl ` R ) | S C_ j } =/= (/) )
10		ssrab2 3687	. . . 4 |- { j e. ( Idl ` R ) | S C_ j } C_ ( Idl ` R )
11		intidl 33828	. . . 4 |- ( ( R e. RingOps /\ { j e. ( Idl ` R ) | S C_ j } =/= (/) /\ { j e. ( Idl ` R ) | S C_ j } C_ ( Idl ` R ) ) -> |^| { j e. ( Idl ` R ) | S C_ j } e. ( Idl ` R ) )
12	10, 11	mp3an3 1413	. . 3 |- ( ( R e. RingOps /\ { j e. ( Idl ` R ) | S C_ j } =/= (/) ) -> |^| { j e. ( Idl ` R ) | S C_ j } e. ( Idl ` R ) )
13	9, 12	syldan 487	. 2 |- ( ( R e. RingOps /\ S C_ X ) -> |^| { j e. ( Idl ` R ) | S C_ j } e. ( Idl ` R ) )
14	3, 13	eqeltrd 2701	1 |- ( ( R e. RingOps /\ S C_ X ) -> ( R IdlGen S ) e. ( Idl ` R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   |^|cint 4475   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   RingOpscrngo 33693   Idlcidl 33806   IdlGen cigen 33858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-rngo 33694  df-idl 33809  df-igen 33859

This theorem is referenced by:  igenval2  33865  isfldidl  33867  ispridlc  33869