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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 ( ). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
ndmaov.1
ndmaovcl.2 (())
ndmaovcl.3 (())
Assertion
Ref Expression
ndmaovcl (())

Proof of Theorem ndmaovcl
StepHypRef Expression
1 ndmaovcl.2 . 2 (())
2 opelxp 4719 . . 3
3 ndmaov.1 . . . . . 6
43eqcomi 2287 . . . . 5
54eleq2i 2347 . . . 4
6 ndmaovcl.3 . . . . 5 (())
7 ndmaov 28043 . . . . 5 (())
8 eleq1 2343 . . . . . . 7 (()) (())
98biimpd 198 . . . . . 6 (()) (())
10 vprc 4152 . . . . . . 7
1110pm2.21i 123 . . . . . 6 (())
129, 11syl6com 31 . . . . 5 (()) (()) (())
136, 7, 12mpsyl 59 . . . 4 (())
145, 13sylnbi 297 . . 3 (())
152, 14sylnbir 298 . 2 (())
161, 15pm2.61i 156 1 (())
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wceq 1623   wcel 1684  cvv 2788  cop 3643   cxp 4687   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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