Theorem rimrcl 14576

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 14569	. 2 |- RingIso = ( r e. _V , s e. _V |-> { f e. ( r RingHom s ) | `' f e. ( s RingHom r ) } )
2	1	elmpocl 6287	1 |- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   {crab 2532   _Vcvv 2821   `'ccnv 4775  (class class class)co 6088   RingHom crh 14566   RingIso crs 14567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-rim 14569

This theorem is used by:  isrim0  14577