Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndmaovcl Unicode version

Theorem ndmaovcl 28171
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6021 where it is required that the domain contains the empty set ( (/) 
e.  S). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
ndmaov.1  |-  dom  F  =  ( S  X.  S )
ndmaovcl.2  |-  ( ( A  e.  S  /\  B  e.  S )  -> (( A F B))  e.  S )
ndmaovcl.3  |- (( A F B))  e.  _V
Assertion
Ref Expression
ndmaovcl  |- (( A F B))  e.  S

Proof of Theorem ndmaovcl
StepHypRef Expression
1 ndmaovcl.2 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  -> (( A F B))  e.  S )
2 opelxp 4735 . . 3  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
3 ndmaov.1 . . . . . 6  |-  dom  F  =  ( S  X.  S )
43eqcomi 2300 . . . . 5  |-  ( S  X.  S )  =  dom  F
54eleq2i 2360 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  dom  F )
6 ndmaovcl.3 . . . . 5  |- (( A F B))  e.  _V
7 ndmaov 28151 . . . . 5  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
8 eleq1 2356 . . . . . . 7  |-  ( (( A F B))  =  _V  ->  ( (( A F B))  e.  _V  <->  _V  e.  _V ) )
98biimpd 198 . . . . . 6  |-  ( (( A F B))  =  _V  ->  ( (( A F B))  e.  _V  ->  _V  e.  _V ) )
10 vprc 4168 . . . . . . 7  |-  -.  _V  e.  _V
1110pm2.21i 123 . . . . . 6  |-  ( _V  e.  _V  -> (( A F B))  e.  S )
129, 11syl6com 31 . . . . 5  |-  ( (( A F B))  e.  _V  ->  ( (( A F B))  =  _V  -> (( A F B))  e.  S ) )
136, 7, 12mpsyl 59 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  e.  S )
145, 13sylnbi 297 . . 3  |-  ( -. 
<. A ,  B >.  e.  ( S  X.  S
)  -> (( A F B))  e.  S )
152, 14sylnbir 298 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  e.  S )
161, 15pm2.61i 156 1  |- (( A F B))  e.  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   dom cdm 4705   ((caov 28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
  Copyright terms: Public domain W3C validator