ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fovcl Unicode version

Theorem fovcl 6025
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
Hypothesis
Ref Expression
fovcl.1  |-  F :
( R  X.  S
) --> C
Assertion
Ref Expression
fovcl  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )

Proof of Theorem fovcl
StepHypRef Expression
1 fovcl.1 . . . 4  |-  F :
( R  X.  S
) --> C
21a1i 9 . . 3  |-  ( A  e.  R  ->  F : ( R  X.  S ) --> C )
32fovcld 6024 . 2  |-  ( ( A  e.  R  /\  A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
433anidm12 1306 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2164    X. cxp 4658   -->wf 5251  (class class class)co 5919
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922
This theorem is referenced by:  xaddcl  9929  ixxssxr  9969  fzof  10213  elfzoelz  10216  fzoval  10217  addcncntoplem  14740
  Copyright terms: Public domain W3C validator