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Theorem ndmovrcl 6718
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 6716 . . . 4 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
43eleq1d 2672 . . 3 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
51, 4mtbiri 316 . 2 (¬ (𝐴𝑆𝐵𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆)
65con4i 112 1 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  c0 3874   × cxp 5036  dom cdm 5038  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  ndmovass  6720  ndmovdistr  6721  ndmovord  6722  ndmovordi  6723  caovmo  6769  brecop2  7728  eceqoveq  7740  addcanpi  9600  mulcanpi  9601  ordpipq  9643  recmulnq  9665  recclnq  9667  ltexnq  9676  nsmallnq  9678  ltbtwnnq  9679  prlem934  9734  ltaddpr  9735  ltaddpr2  9736  ltexprlem2  9738  ltexprlem3  9739  ltexprlem4  9740  ltexprlem6  9742  ltexprlem7  9743  addcanpr  9747  prlem936  9748  mappsrpr  9808
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