Theorem maxidlval 33838

Description: The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)

Hypotheses

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

Assertion

Ref	Expression
maxidlval	|- ( R e. RingOps -> ( MaxIdl ` R ) = { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } )

Distinct variable group:   R, i, j

Allowed substitution hints:   G( i, j)   X( i, j)

Proof of Theorem maxidlval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( r = R -> ( Idl ` r ) = ( Idl ` R ) )
2		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		maxidlval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5353	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		maxidlval.2	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8	7	neeq2d 2854	. . . 4 |- ( r = R -> ( i =/= ran ( 1st ` r ) <-> i =/= X ) )
9	7	eqeq2d 2632	. . . . . . 7 |- ( r = R -> ( j = ran ( 1st ` r ) <-> j = X ) )
10	9	orbi2d 738	. . . . . 6 |- ( r = R -> ( ( j = i \/ j = ran ( 1st ` r ) ) <-> ( j = i \/ j = X ) ) )
11	10	imbi2d 330	. . . . 5 |- ( r = R -> ( ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) <-> ( i C_ j -> ( j = i \/ j = X ) ) ) )
12	1, 11	raleqbidv 3152	. . . 4 |- ( r = R -> ( A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) <-> A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) )
13	8, 12	anbi12d 747	. . 3 |- ( r = R -> ( ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) <-> ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) ) )
14	1, 13	rabeqbidv 3195	. 2 |- ( r = R -> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } = { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } )
15		df-maxidl 33811	. 2 |- MaxIdl = ( r e. RingOps |-> { i e. ( Idl ` r ) | ( i =/= ran ( 1st ` r ) /\ A. j e. ( Idl ` r ) ( i C_ j -> ( j = i \/ j = ran ( 1st ` r ) ) ) ) } )
16		fvex 6201	. . 3 |- ( Idl ` R ) e. _V
17	16	rabex 4813	. 2 |- { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } e. _V
18	14, 15, 17	fvmpt 6282	1 |- ( R e. RingOps -> ( MaxIdl ` R ) = { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   ran crn 5115   ` cfv 5888   1stc1st 7166   RingOpscrngo 33693   Idlcidl 33806   MaxIdlcmaxidl 33808

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fv 5896  df-maxidl 33811

This theorem is referenced by:  ismaxidl  33839