Theorem isrim0 14038

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 14037	. 2 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
2		rhmrcl1 14032	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	2	elexd 2790	. . . 4 |- ( F e. ( R RingHom S ) -> R e. _V )
4		rhmrcl2 14033	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
5	4	elexd 2790	. . . 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 14030	. . . . . 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 5976	. . . . . . 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 5976	. . . . . . . . 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 2277	. . . . . 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 2771	. . . . 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 14034	. . . . . . 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 4203	. . . . . 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 6096	. . . 4 |- ( ( R e. _V /\ S e. _V ) -> ( R RingIso S ) = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
24	23	eleq2d 2277	. . 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 4870	. . . . 5 |- ( f = F -> `' f = `' F )
26	25	eleq1d 2276	. . . 4 |- ( f = F -> ( `' f e. ( S RingHom R ) <-> `' F e. ( S RingHom R ) ) )
27	26	elrab 2936	. . 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 706	1 |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {crab 2490   _Vcvv 2776   `'ccnv 4692  (class class class)co 5967   e. cmpo 5969   Ringcrg 13873   RingHom crh 14027   RingIso crs 14028

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-mhm 13406  df-ghm 13692  df-mgp 13798  df-ur 13837  df-ring 13875  df-rhm 14029  df-rim 14030

This theorem is referenced by:  isrim  14046  rimrhm  14048