Theorem isrim0 14441

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 14440	. 2 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
2		rhmrcl1 14435	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	2	elexd 2835	. . . 4 |- ( F e. ( R RingHom S ) -> R e. _V )
4		rhmrcl2 14436	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
5	4	elexd 2835	. . . 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 14433	. . . . . 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 6084	. . . . . . 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 6084	. . . . . . . . 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 2308	. . . . . 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 2816	. . . . 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 14437	. . . . . . 7 |- ( ( R e. _V /\ S e. _V ) -> ( R RingHom S ) e. _V )
20	17, 18, 19	syl2anc 415	. . . . . 6 |- ( ( R e. _V /\ S e. _V ) -> ( R RingHom S ) e. _V )
21		rabexg 4274	. . . . . 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 6206	. . . 4 |- ( ( R e. _V /\ S e. _V ) -> ( R RingIso S ) = { f e. ( R RingHom S ) | `' f e. ( S RingHom R ) } )
24	23	eleq2d 2308	. . 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 4949	. . . . 5 |- ( f = F -> `' f = `' F )
26	25	eleq1d 2307	. . . 4 |- ( f = F -> ( `' f e. ( S RingHom R ) <-> `' F e. ( S RingHom R ) ) )
27	26	elrab 2982	. . 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 716	1 |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   `'ccnv 4768  (class class class)co 6075   e. cmpo 6077   Ringcrg 14274   RingHom crh 14430   RingIso crs 14431

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-mhm 13743  df-ghm 14021  df-mgp 14195  df-ur 14238  df-ring 14276  df-rhm 14432  df-rim 14433

This theorem is referenced by:  isrim  14449  rimrhm  14451