Theorem brric 18744

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)

Assertion

Ref	Expression
brric	|- ( R ~=r S <-> ( R RingIso S ) =/= (/) )

Proof of Theorem brric

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

Step	Hyp	Ref	Expression
1		df-ric 18718	. 2 |- ~=r = ( `' RingIso " ( _V \ 1o ) )
2		ovex 6678	. . . . 5 |- ( r RingHom s ) e. _V
3		rabexg 4812	. . . . 5 |- ( ( r RingHom s ) e. _V -> { h e. ( r RingHom s ) | `' h e. ( s RingHom r ) } e. _V )
4	2, 3	mp1i 13	. . . 4 |- ( ( r e. _V /\ s e. _V ) -> { h e. ( r RingHom s ) | `' h e. ( s RingHom r ) } e. _V )
5	4	rgen2a 2977	. . 3 |- A. r e. _V A. s e. _V { h e. ( r RingHom s ) | `' h e. ( s RingHom r ) } e. _V
6		df-rngiso 18716	. . . 4 |- RingIso = ( r e. _V , s e. _V |-> { h e. ( r RingHom s ) | `' h e. ( s RingHom r ) } )
7	6	fnmpt2 7238	. . 3 |- ( A. r e. _V A. s e. _V { h e. ( r RingHom s ) | `' h e. ( s RingHom r ) } e. _V -> RingIso Fn ( _V X. _V ) )
8	5, 7	ax-mp 5	. 2 |- RingIso Fn ( _V X. _V )
9	1, 8	brwitnlem 7587	1 |- ( R ~=r S <-> ( R RingIso S ) =/= (/) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   Fn wfn 5883  (class class class)co 6650   RingHom crh 18712   RingIso crs 18713   ~=r cric 18714

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-rngiso 18716  df-ric 18718

This theorem is referenced by:  brric2  18745  mat1ric  20293  scmatric  20343  matcpmric  20564  pmmpric  20628