Theorem rngcrescrhmALTV 42103

Description: The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rngcrescrhmALTV.u	|- ( ph -> U e. V )
rngcrescrhmALTV.c	|- C = (RngCatALTV ` U )
rngcrescrhmALTV.r	|- ( ph -> R = ( Ring i^i U ) )
rngcrescrhmALTV.h	|- H = ( RingHom |` ( R X. R ) )

Assertion

Ref	Expression
rngcrescrhmALTV	|- ( ph -> ( C |`cat H ) = ( ( C ↾s R ) sSet <. ( Hom ` ndx ) , H >. ) )

Proof of Theorem rngcrescrhmALTV

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( C |`cat H ) = ( C |`cat H )
2		rngcrescrhmALTV.c	. . . 4 |- C = (RngCatALTV ` U )
3		fvex 6201	. . . 4 |- (RngCatALTV ` U ) e. _V
4	2, 3	eqeltri 2697	. . 3 |- C e. _V
5	4	a1i 11	. 2 |- ( ph -> C e. _V )
6		rngcrescrhmALTV.r	. . . 4 |- ( ph -> R = ( Ring i^i U ) )
7		incom 3805	. . . 4 |- ( Ring i^i U ) = ( U i^i Ring )
8	6, 7	syl6eq 2672	. . 3 |- ( ph -> R = ( U i^i Ring ) )
9		rngcrescrhmALTV.u	. . . 4 |- ( ph -> U e. V )
10		inex1g 4801	. . . 4 |- ( U e. V -> ( U i^i Ring ) e. _V )
11	9, 10	syl 17	. . 3 |- ( ph -> ( U i^i Ring ) e. _V )
12	8, 11	eqeltrd 2701	. 2 |- ( ph -> R e. _V )
13		inss1 3833	. . . . . 6 |- ( Ring i^i U ) C_ Ring
14	6, 13	syl6eqss 3655	. . . . 5 |- ( ph -> R C_ Ring )
15		xpss12 5225	. . . . 5 |- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) )
16	14, 14, 15	syl2anc 693	. . . 4 |- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) )
17		rhmfn 41918	. . . . 5 |- RingHom Fn ( Ring X. Ring )
18		fnssresb 6003	. . . . 5 |- ( RingHom Fn ( Ring X. Ring ) -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) )
19	17, 18	mp1i 13	. . . 4 |- ( ph -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) )
20	16, 19	mpbird 247	. . 3 |- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) )
21		rngcrescrhmALTV.h	. . . 4 |- H = ( RingHom |` ( R X. R ) )
22	21	fneq1i 5985	. . 3 |- ( H Fn ( R X. R ) <-> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) )
23	20, 22	sylibr 224	. 2 |- ( ph -> H Fn ( R X. R ) )
24	1, 5, 12, 23	rescval2 16488	1 |- ( ph -> ( C |`cat H ) = ( ( C ↾s R ) sSet <. ( Hom ` ndx ) , H >. ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   X. cxp 5112   |` cres 5116   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   ↾_s cress 15858   Hom chom 15952   |`_cat cresc 16468   Ringcrg 18547   RingHom crh 18712  RngCatALTVcrngcALTV 41958

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  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-resc 16471  df-mhm 17335  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715

This theorem is referenced by: (None)