Theorem rhmima 14200

Description: The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
rhmima	|- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubRing ` N ) )

Proof of Theorem rhmima

Step	Hyp	Ref	Expression
1		rhmghm 14111	. . 3 |- ( F e. ( M RingHom N ) -> F e. ( M GrpHom N ) )
2		subrgsubg 14176	. . 3 |- ( X e. (SubRing ` M ) -> X e. (SubGrp ` M ) )
3		ghmima 13788	. . 3 |- ( ( F e. ( M GrpHom N ) /\ X e. (SubGrp ` M ) ) -> ( F " X ) e. (SubGrp ` N ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubGrp ` N ) )
5		eqid 2229	. . . 4 |- (mulGrp ` M ) = (mulGrp ` M )
6		eqid 2229	. . . 4 |- (mulGrp ` N ) = (mulGrp ` N )
7	5, 6	rhmmhm 14108	. . 3 |- ( F e. ( M RingHom N ) -> F e. ( (mulGrp ` M ) MndHom (mulGrp ` N ) ) )
8	5	subrgsubm 14183	. . 3 |- ( X e. (SubRing ` M ) -> X e. (SubMnd ` (mulGrp ` M ) ) )
9		mhmima 13510	. . 3 |- ( ( F e. ( (mulGrp ` M ) MndHom (mulGrp ` N ) ) /\ X e. (SubMnd ` (mulGrp ` M ) ) ) -> ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) )
10	7, 8, 9	syl2an 289	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) )
11		rhmrcl2 14105	. . . 4 |- ( F e. ( M RingHom N ) -> N e. Ring )
12	11	adantr 276	. . 3 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> N e. Ring )
13	6	issubrg3 14196	. . 3 |- ( N e. Ring -> ( ( F " X ) e. (SubRing ` N ) <-> ( ( F " X ) e. (SubGrp ` N ) /\ ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) ) ) )
14	12, 13	syl 14	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( ( F " X ) e. (SubRing ` N ) <-> ( ( F " X ) e. (SubGrp ` N ) /\ ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) ) ) )
15	4, 10, 14	mpbir2and 950	1 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubRing ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   "cima 4719   ` cfv 5314  (class class class)co 5994   MndHom cmhm 13476  SubMndcsubmnd 13477  SubGrpcsubg 13690   GrpHom cghm 13763  mulGrpcmgp 13869   Ringcrg 13945   RingHom crh 14099  SubRingcsubrg 14166

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-map 6787  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-iress 13026  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-mhm 13478  df-submnd 13479  df-grp 13522  df-minusg 13523  df-subg 13693  df-ghm 13764  df-mgp 13870  df-ur 13909  df-ring 13947  df-rhm 14101  df-subrg 14168

This theorem is referenced by:  rnrhmsubrg  14201