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Theorem ovrcl 6830
 Description: Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
Hypothesis
Ref Expression
ovprc1.1 Rel dom 𝐹
Assertion
Ref Expression
ovrcl (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem ovrcl
StepHypRef Expression
1 n0i 4066 . 2 (𝐶 ∈ (𝐴𝐹𝐵) → ¬ (𝐴𝐹𝐵) = ∅)
2 ovprc1.1 . . 3 Rel dom 𝐹
32ovprc 6827 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
41, 3nsyl2 144 1 (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ∅c0 4061  dom cdm 5249  Rel wrel 5254  (class class class)co 6792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6795 This theorem is referenced by:  cda1dif  9199  smatrcl  30196
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