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Theorem ndmovrcl 7323
Description: Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.)
Hypotheses
Ref Expression
ndmov.1 dom 𝐹 = (𝑆 × 𝑆)
ndmovrcl.3 ¬ ∅ ∈ 𝑆
Assertion
Ref Expression
ndmovrcl ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))

Proof of Theorem ndmovrcl
StepHypRef Expression
1 ndmovrcl.3 . . 3 ¬ ∅ ∈ 𝑆
2 ndmov.1 . . . . 5 dom 𝐹 = (𝑆 × 𝑆)
32ndmov 7321 . . . 4 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
43eleq1d 2894 . . 3 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
51, 4mtbiri 328 . 2 (¬ (𝐴𝑆𝐵𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆)
65con4i 114 1 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  c0 4288   × cxp 5546  dom cdm 5548  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  ndmovass  7325  ndmovdistr  7326  ndmovord  7327  ndmovordi  7328  caovmo  7374  brecop2  8380  eceqoveq  8391  addcanpi  10309  mulcanpi  10310  ordpipq  10352  recmulnq  10374  recclnq  10376  ltexnq  10385  nsmallnq  10387  ltbtwnnq  10388  prlem934  10443  ltaddpr  10444  ltaddpr2  10445  ltexprlem2  10447  ltexprlem3  10448  ltexprlem4  10449  ltexprlem6  10451  ltexprlem7  10452  addcanpr  10456  prlem936  10457  mappsrpr  10518
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