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Theorem ndmovrcl 7607
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 7605 . . . 4 (¬ (𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) = ∅)
43eleq1d 2810 . . 3 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
51, 4mtbiri 326 . 2 (¬ (𝐴𝑆𝐵𝑆) → ¬ (𝐴𝐹𝐵) ∈ 𝑆)
65con4i 114 1 ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  c0 4322   × cxp 5676  dom cdm 5678  (class class class)co 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-dm 5688  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  ndmovass  7609  ndmovdistr  7610  ndmovord  7611  ndmovordi  7612  caovmo  7658  brecop2  8830  eceqoveq  8841  addcanpi  10924  mulcanpi  10925  ordpipq  10967  recmulnq  10989  recclnq  10991  ltexnq  11000  nsmallnq  11002  ltbtwnnq  11003  prlem934  11058  ltaddpr  11059  ltaddpr2  11060  ltexprlem2  11062  ltexprlem3  11063  ltexprlem4  11064  ltexprlem6  11066  ltexprlem7  11067  addcanpr  11071  prlem936  11072  mappsrpr  11133
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