Theorem rnrhmsubrg 14483

Description: The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)

Assertion

Ref	Expression
rnrhmsubrg	|- ( F e. ( M RingHom N ) -> ran F e. (SubRing ` N ) )

Proof of Theorem rnrhmsubrg

Step	Hyp	Ref	Expression
1		df-ima 4767	. . 3 |- ( F " ( Base ` M ) ) = ran ( F |` ( Base ` M ) )
2		eqid 2234	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
3		eqid 2234	. . . . . . 7 |- ( Base ` N ) = ( Base ` N )
4	2, 3	rhmf 14393	. . . . . 6 |- ( F e. ( M RingHom N ) -> F : ( Base ` M ) --> ( Base ` N ) )
5	4	ffnd 5514	. . . . 5 |- ( F e. ( M RingHom N ) -> F Fn ( Base ` M ) )
6		fnresdm 5472	. . . . 5 |- ( F Fn ( Base ` M ) -> ( F |` ( Base ` M ) ) = F )
7	5, 6	syl 14	. . . 4 |- ( F e. ( M RingHom N ) -> ( F |` ( Base ` M ) ) = F )
8	7	rneqd 4991	. . 3 |- ( F e. ( M RingHom N ) -> ran ( F |` ( Base ` M ) ) = ran F )
9	1, 8	eqtr2id 2280	. 2 |- ( F e. ( M RingHom N ) -> ran F = ( F " ( Base ` M ) ) )
10		rhmrcl1 14385	. . . 4 |- ( F e. ( M RingHom N ) -> M e. Ring )
11	2	subrgid 14454	. . . 4 |- ( M e. Ring -> ( Base ` M ) e. (SubRing ` M ) )
12	10, 11	syl 14	. . 3 |- ( F e. ( M RingHom N ) -> ( Base ` M ) e. (SubRing ` M ) )
13		rhmima 14482	. . 3 |- ( ( F e. ( M RingHom N ) /\ ( Base ` M ) e. (SubRing ` M ) ) -> ( F " ( Base ` M ) ) e. (SubRing ` N ) )
14	12, 13	mpdan 421	. 2 |- ( F e. ( M RingHom N ) -> ( F " ( Base ` M ) ) e. (SubRing ` N ) )
15	9, 14	eqeltrd 2311	1 |- ( F e. ( M RingHom N ) -> ran F e. (SubRing ` N ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ran crn 4755   |` cres 4756   "cima 4757   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   Ringcrg 14224   RingHom crh 14380  SubRingcsubrg 14448

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-mhm 13756  df-submnd 13757  df-grp 13800  df-minusg 13801  df-subg 13971  df-ghm 14042  df-mgp 14149  df-ur 14188  df-ring 14226  df-rhm 14382  df-subrg 14450

This theorem is referenced by: (None)