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Theorem ndmovcl 6172
Description: The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.)
Hypotheses
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
ndmovcl.2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
ndmovcl.3  |-  (/)  e.  S
Assertion
Ref Expression
ndmovcl  |-  ( A F B )  e.  S

Proof of Theorem ndmovcl
StepHypRef Expression
1 ndmovcl.2 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
2 ndmov.1 . . . 4  |-  dom  F  =  ( S  X.  S )
32ndmov 6171 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
4 ndmovcl.3 . . 3  |-  (/)  e.  S
53, 4syl6eqel 2476 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  e.  S
)
61, 5pm2.61i 158 1  |-  ( A F B )  e.  S
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   (/)c0 3572    X. cxp 4817   dom cdm 4819  (class class class)co 6021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-dm 4829  df-iota 5359  df-fv 5403  df-ov 6024
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