Theorem isrim0 14174

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 14173	. 2 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
2		rhmrcl1 14168	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	2	elexd 2816	. . . 4 |- ( F e. ( R RingHom S ) -> R e. _V )
4		rhmrcl2 14169	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
5	4	elexd 2816	. . . 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 14166	. . . . . 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 6026	. . . . . . 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 6026	. . . . . . . . 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 2301	. . . . . 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 2797	. . . . 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 14170	. . . . . . 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 4233	. . . . . 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 6148	. . . 4 |- ( ( R e. _V /\ S e. _V ) -> ( R RingIso S ) = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
24	23	eleq2d 2301	. . 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 4904	. . . . 5 |- ( f = F -> `' f = `' F )
26	25	eleq1d 2300	. . . 4 |- ( f = F -> ( `' f e. ( S RingHom R ) <-> `' F e. ( S RingHom R ) ) )
27	26	elrab 2962	. . 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 711	1 |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   `'ccnv 4724  (class class class)co 6017   e. cmpo 6019   Ringcrg 14008   RingHom crh 14163   RingIso crs 14164

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-mhm 13541  df-ghm 13827  df-mgp 13933  df-ur 13972  df-ring 14010  df-rhm 14165  df-rim 14166

This theorem is referenced by:  isrim  14182  rimrhm  14184