Theorem isrim0 13717

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 13716	. 2 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
2		rhmrcl1 13711	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	2	elexd 2776	. . . 4 |- ( F e. ( R RingHom S ) -> R e. _V )
4		rhmrcl2 13712	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
5	4	elexd 2776	. . . 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 13709	. . . . . 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 5931	. . . . . . 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 5931	. . . . . . . . 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 2266	. . . . . 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 2758	. . . . 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 13713	. . . . . . 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 4176	. . . . . 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 6050	. . . 4 |- ( ( R e. _V /\ S e. _V ) -> ( R RingIso S ) = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
24	23	eleq2d 2266	. . 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 4840	. . . . 5 |- ( f = F -> `' f = `' F )
26	25	eleq1d 2265	. . . 4 |- ( f = F -> ( `' f e. ( S RingHom R ) <-> `' F e. ( S RingHom R ) ) )
27	26	elrab 2920	. . 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 705	1 |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   {crab 2479   _Vcvv 2763   `'ccnv 4662  (class class class)co 5922   e. cmpo 5924   Ringcrg 13552   RingHom crh 13706   RingIso crs 13707

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-mhm 13091  df-ghm 13371  df-mgp 13477  df-ur 13516  df-ring 13554  df-rhm 13708  df-rim 13709

This theorem is referenced by:  isrim  13725  rimrhm  13727