Theorem isrngisom 41896

Description: An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)

Assertion

Ref	Expression
isrngisom	|- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngIsom S ) <-> ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) ) )

Proof of Theorem isrngisom

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

Step	Hyp	Ref	Expression
1		df-rngisom 41888	. . . . 5 |- RngIsom = ( r e. _V , s e. _V |-> { f e. ( r RngHomo s ) | `' f e. ( s RngHomo r ) } )
2	1	a1i 11	. . . 4 |- ( ( R e. V /\ S e. W ) -> RngIsom = ( r e. _V , s e. _V |-> { f e. ( r RngHomo s ) | `' f e. ( s RngHomo r ) } ) )
3		oveq12 6659	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( r RngHomo s ) = ( R RngHomo S ) )
4	3	adantl 482	. . . . 5 |- ( ( ( R e. V /\ S e. W ) /\ ( r = R /\ s = S ) ) -> ( r RngHomo s ) = ( R RngHomo S ) )
5		oveq12 6659	. . . . . . . 8 |- ( ( s = S /\ r = R ) -> ( s RngHomo r ) = ( S RngHomo R ) )
6	5	ancoms 469	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( s RngHomo r ) = ( S RngHomo R ) )
7	6	adantl 482	. . . . . 6 |- ( ( ( R e. V /\ S e. W ) /\ ( r = R /\ s = S ) ) -> ( s RngHomo r ) = ( S RngHomo R ) )
8	7	eleq2d 2687	. . . . 5 |- ( ( ( R e. V /\ S e. W ) /\ ( r = R /\ s = S ) ) -> ( `' f e. ( s RngHomo r ) <-> `' f e. ( S RngHomo R ) ) )
9	4, 8	rabeqbidv 3195	. . . 4 |- ( ( ( R e. V /\ S e. W ) /\ ( r = R /\ s = S ) ) -> { f e. ( r RngHomo s ) | `' f e. ( s RngHomo r ) } = { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } )
10		elex 3212	. . . . 5 |- ( R e. V -> R e. _V )
11	10	adantr 481	. . . 4 |- ( ( R e. V /\ S e. W ) -> R e. _V )
12		elex 3212	. . . . 5 |- ( S e. W -> S e. _V )
13	12	adantl 482	. . . 4 |- ( ( R e. V /\ S e. W ) -> S e. _V )
14		ovex 6678	. . . . . 6 |- ( R RngHomo S ) e. _V
15	14	rabex 4813	. . . . 5 |- { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } e. _V
16	15	a1i 11	. . . 4 |- ( ( R e. V /\ S e. W ) -> { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } e. _V )
17	2, 9, 11, 13, 16	ovmpt2d 6788	. . 3 |- ( ( R e. V /\ S e. W ) -> ( R RngIsom S ) = { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } )
18	17	eleq2d 2687	. 2 |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngIsom S ) <-> F e. { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } ) )
19		cnveq 5296	. . . 4 |- ( f = F -> `' f = `' F )
20	19	eleq1d 2686	. . 3 |- ( f = F -> ( `' f e. ( S RngHomo R ) <-> `' F e. ( S RngHomo R ) ) )
21	20	elrab 3363	. 2 |- ( F e. { f e. ( R RngHomo S ) | `' f e. ( S RngHomo R ) } <-> ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) )
22	18, 21	syl6bb 276	1 |- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngIsom S ) <-> ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   `'ccnv 5113  (class class class)co 6650   |-> cmpt2 6652   RngHomo crngh 41885   RngIsom crngs 41886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rngisom 41888

This theorem is referenced by:  isrngim  41904  rngcinv  41981  rngcinvALTV  41993