Theorem rimrcl 14118

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 14111	. 2 |- RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } )
2	1	elmpocl 6199	1 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   {crab 2512   _Vcvv 2799   `'ccnv 4717  (class class class)co 6000   RingHom crh 14108   RingIso crs 14109

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 4201  ax-pow 4257  ax-pr 4292

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-rim 14111

This theorem is referenced by:  isrim0  14119