Theorem rimrcl 14164

Description: Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)

Assertion

Ref	Expression
rimrcl	|- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )

Proof of Theorem rimrcl

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

Step	Hyp	Ref	Expression
1		df-rim 14157	. 2 |- RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } )
2	1	elmpocl 6212	1 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   {crab 2512   _Vcvv 2800   `'ccnv 4722  (class class class)co 6013   RingHom crh 14154   RingIso crs 14155

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-rim 14157

This theorem is referenced by:  isrim0  14165