Theorem ressmex 13472

Description: If a structure restriction is inhabited, the structure is a set and so is the class it is restricted to. (Contributed by Jim Kingdon, 16-Sep-2026.)

Hypothesis

Ref	Expression
ressmex.r	|- R = ( W ↾s A )

Assertion

Ref	Expression
ressmex	|- ( X e. R -> ( W e. _V /\ A e. _V ) )

Proof of Theorem ressmex

Dummy variables x w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iress 13412	. . 3 |- ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
2	1	elmpocl 6284	. 2 |- ( X e. ( W ↾s A ) -> ( W e. _V /\ A e. _V ) )
3		ressmex.r	. 2 |- R = ( W ↾s A )
4	2, 3	eleq2s 2333	1 |- ( X e. R -> ( W e. _V /\ A e. _V ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   <.cop 3712   ` cfv 5377  (class class class)co 6085   ndxcnx 13401   sSet csts 13402   Basecbs 13404   ↾_s cress 13405

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 4249  ax-pow 4311  ax-pr 4346

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 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-iress 13412

This theorem is used by:  resscntz  14160