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Theorem ndmaovcl 28035
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6225 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 4901 . . 3  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
3 ndmaov.1 . . . . . 6  |-  dom  F  =  ( S  X.  S )
43eqcomi 2440 . . . . 5  |-  ( S  X.  S )  =  dom  F
54eleq2i 2500 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  dom  F )
6 ndmaovcl.3 . . . . 5  |- (( A F B))  e.  _V
7 ndmaov 28015 . . . . 5  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
8 eleq1 2496 . . . . . . 7  |-  ( (( A F B))  =  _V  ->  ( (( A F B))  e.  _V  <->  _V  e.  _V ) )
98biimpd 199 . . . . . 6  |-  ( (( A F B))  =  _V  ->  ( (( A F B))  e.  _V  ->  _V  e.  _V ) )
10 vprc 4334 . . . . . . 7  |-  -.  _V  e.  _V
1110pm2.21i 125 . . . . . 6  |-  ( _V  e.  _V  -> (( A F B))  e.  S )
129, 11syl6com 33 . . . . 5  |-  ( (( A F B))  e.  _V  ->  ( (( A F B))  =  _V  -> (( A F B))  e.  S ) )
136, 7, 12mpsyl 61 . . . 4  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  e.  S )
145, 13sylnbi 298 . . 3  |-  ( -. 
<. A ,  B >.  e.  ( S  X.  S
)  -> (( A F B))  e.  S )
152, 14sylnbir 299 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  e.  S )
161, 15pm2.61i 158 1  |- (( A F B))  e.  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2949   <.cop 3810    X. cxp 4869   dom cdm 4871   ((caov 27941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-opab 4260  df-xp 4877  df-fv 5455  df-dfat 27942  df-afv 27943  df-aov 27944
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