Theorem isrim0 14140

Description: A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)

Assertion

Ref	Expression
isrim0	|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Proof of Theorem isrim0

Dummy variables f r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rimrcl 14139	. 2 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
2		rhmrcl1 14134	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	2	elexd 2813	. . . 4 |- ( F e. ( R RingHom S ) -> R e. _V )
4		rhmrcl2 14135	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
5	4	elexd 2813	. . . 4 |- ( F e. ( R RingHom S ) -> S e. _V )
6	3, 5	jca 306	. . 3 |- ( F e. ( R RingHom S ) -> ( R e. _V /\ S e. _V ) )
7	6	adantr 276	. 2 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> ( R e. _V /\ S e. _V ) )
8		df-rim 14132	. . . . . 6 |- RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } )
9	8	a1i 9	. . . . 5 |- ( ( R e. _V /\ S e. _V ) -> RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } ) )
10		oveq12 6016	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( r RingHom s ) = ( R RingHom S ) )
11	10	adantl 277	. . . . . 6 |- ( ( ( R e. _V /\ S e. _V ) /\ ( r = R /\ s = S ) ) -> ( r RingHom s ) = ( R RingHom S ) )
12		oveq12 6016	. . . . . . . . 9 |- ( ( s = S /\ r = R ) -> ( s RingHom r ) = ( S RingHom R ) )
13	12	ancoms 268	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( s RingHom r ) = ( S RingHom R ) )
14	13	adantl 277	. . . . . . 7 |- ( ( ( R e. _V /\ S e. _V ) /\ ( r = R /\ s = S ) ) -> ( s RingHom r ) = ( S RingHom R ) )
15	14	eleq2d 2299	. . . . . 6 |- ( ( ( R e. _V /\ S e. _V ) /\ ( r = R /\ s = S ) ) -> ( `' f e. ( s RingHom r ) <-> `' f e. ( S RingHom R ) ) )
16	11, 15	rabeqbidv 2794	. . . . 5 |- ( ( ( R e. _V /\ S e. _V ) /\ ( r = R /\ s = S ) ) -> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
17		simpl 109	. . . . 5 |- ( ( R e. _V /\ S e. _V ) -> R e. _V )
18		simpr 110	. . . . 5 |- ( ( R e. _V /\ S e. _V ) -> S e. _V )
19		rhmex 14136	. . . . . . 7 |- ( ( R e. _V /\ S e. _V ) -> ( R RingHom S ) e. _V )
20	17, 18, 19	syl2anc 411	. . . . . 6 |- ( ( R e. _V /\ S e. _V ) -> ( R RingHom S ) e. _V )
21		rabexg 4227	. . . . . 6 |- ( ( R RingHom S ) e. _V -> { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } e. _V )
22	20, 21	syl 14	. . . . 5 |- ( ( R e. _V /\ S e. _V ) -> { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } e. _V )
23	9, 16, 17, 18, 22	ovmpod 6138	. . . 4 |- ( ( R e. _V /\ S e. _V ) -> ( R RingIso S ) = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
24	23	eleq2d 2299	. . 3 |- ( ( R e. _V /\ S e. _V ) -> ( F e. ( R RingIso S ) <-> F e. { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } ) )
25		cnveq 4896	. . . . 5 |- ( f = F -> `' f = `' F )
26	25	eleq1d 2298	. . . 4 |- ( f = F -> ( `' f e. ( S RingHom R ) <-> `' F e. ( S RingHom R ) ) )
27	26	elrab 2959	. . 3 |- ( F e. { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )
28	24, 27	bitrdi 196	. 2 |- ( ( R e. _V /\ S e. _V ) -> ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) ) )
29	1, 7, 28	pm5.21nii 709	1 |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   `'ccnv 4718  (class class class)co 6007   e. cmpo 6009   Ringcrg 13974   RingHom crh 14129   RingIso crs 14130

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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

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-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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-plusg 13138  df-mulr 13139  df-mhm 13507  df-ghm 13793  df-mgp 13899  df-ur 13938  df-ring 13976  df-rhm 14131  df-rim 14132

This theorem is referenced by:  isrim  14148  rimrhm  14150