Theorem cntzmhm2 14168

Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
cntzmhm.z	|- Z = (Cntz ` G )
cntzmhm.y	|- Y = (Cntz ` H )

Assertion

Ref	Expression
cntzmhm2	|- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> ( F " S ) C_ ( Y ` ( F " T ) ) )

Proof of Theorem cntzmhm2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzmhm.z	. . . . 5 |- Z = (Cntz ` G )
2		cntzmhm.y	. . . . 5 |- Y = (Cntz ` H )
3	1, 2	cntzmhm 14167	. . . 4 |- ( ( F e. ( G MndHom H ) /\ x e. ( Z ` T ) ) -> ( F ` x ) e. ( Y ` ( F " T ) ) )
4	3	ralrimiva 2623	. . 3 |- ( F e. ( G MndHom H ) -> A. x e. ( Z ` T ) ( F ` x ) e. ( Y ` ( F " T ) ) )
5		ssralv 3312	. . 3 |- ( S C_ ( Z ` T ) -> ( A. x e. ( Z ` T ) ( F ` x ) e. ( Y ` ( F " T ) ) -> A. x e. S ( F ` x ) e. ( Y ` ( F " T ) ) ) )
6	4, 5	mpan9 281	. 2 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> A. x e. S ( F ` x ) e. ( Y ` ( F " T ) ) )
7		eqid 2238	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
8		eqid 2238	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
9	7, 8	mhmf 13825	. . . . 5 |- ( F e. ( G MndHom H ) -> F : ( Base ` G ) --> ( Base ` H ) )
10	9	adantr 276	. . . 4 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> F : ( Base ` G ) --> ( Base ` H ) )
11	10	ffund 5537	. . 3 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> Fun F )
12		simpr 110	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> S C_ ( Z ` T ) )
13	7, 1	cntzssv 14154	. . . . 5 |- ( Z ` T ) C_ ( Base ` G )
14	12, 13	sstrdi 3260	. . . 4 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> S C_ ( Base ` G ) )
15	10	fdmd 5540	. . . 4 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> dom F = ( Base ` G ) )
16	14, 15	sseqtrrd 3287	. . 3 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> S C_ dom F )
17		funimass4 5753	. . 3 |- ( ( Fun F /\ S C_ dom F ) -> ( ( F " S ) C_ ( Y ` ( F " T ) ) <-> A. x e. S ( F ` x ) e. ( Y ` ( F " T ) ) ) )
18	11, 16, 17	syl2anc 415	. 2 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> ( ( F " S ) C_ ( Y ` ( F " T ) ) <-> A. x e. S ( F ` x ) e. ( Y ` ( F " T ) ) ) )
19	6, 18	mpbird 167	1 |- ( ( F e. ( G MndHom H ) /\ S C_ ( Z ` T ) ) -> ( F " S ) C_ ( Y ` ( F " T ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   dom cdm 4774   "cima 4777   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   Basecbs 13404   MndHom cmhm 13817  Cntzccntz 14140

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-mhm 13819  df-cntz 14142

This theorem is used by: (None)