Theorem elcntzsn 17758

Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
cntzfval.b	|- B = ( Base ` M )
cntzfval.p	|- .+ = ( +g ` M )
cntzfval.z	|- Z = (Cntz ` M )

Assertion

Ref	Expression
elcntzsn	|- ( Y e. B -> ( X e. ( Z ` { Y } ) <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) ) )

Proof of Theorem elcntzsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzfval.b	. . . 4 |- B = ( Base ` M )
2		cntzfval.p	. . . 4 |- .+ = ( +g ` M )
3		cntzfval.z	. . . 4 |- Z = (Cntz ` M )
4	1, 2, 3	cntzsnval 17757	. . 3 |- ( Y e. B -> ( Z ` { Y } ) = { x e. B | ( x .+ Y ) = ( Y .+ x ) } )
5	4	eleq2d 2687	. 2 |- ( Y e. B -> ( X e. ( Z ` { Y } ) <-> X e. { x e. B | ( x .+ Y ) = ( Y .+ x ) } ) )
6		oveq1 6657	. . . 4 |- ( x = X -> ( x .+ Y ) = ( X .+ Y ) )
7		oveq2 6658	. . . 4 |- ( x = X -> ( Y .+ x ) = ( Y .+ X ) )
8	6, 7	eqeq12d 2637	. . 3 |- ( x = X -> ( ( x .+ Y ) = ( Y .+ x ) <-> ( X .+ Y ) = ( Y .+ X ) ) )
9	8	elrab 3363	. 2 |- ( X e. { x e. B | ( x .+ Y ) = ( Y .+ x ) } <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) )
10	5, 9	syl6bb 276	1 |- ( Y e. B -> ( X e. ( Z ` { Y } ) <-> ( X e. B /\ ( X .+ Y ) = ( Y .+ X ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Cntzccntz 17748

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-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-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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-cntz 17750

This theorem is referenced by:  gsumconst  18334  gsumpt  18361