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Theorem ndmaovcl 28063
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6005 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 4719 . . 3  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
3 ndmaov.1 . . . . . 6  |-  dom  F  =  ( S  X.  S )
43eqcomi 2287 . . . . 5  |-  ( S  X.  S )  =  dom  F
54eleq2i 2347 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  dom  F )
6 ndmaovcl.3 . . . . 5  |- (( A F B))  e.  _V
7 ndmaov 28043 . . . . 5  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
8 eleq1 2343 . . . . . . 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 4152 . . . . . . 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 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   dom cdm 4689   ((caov 27973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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