Theorem ovrcl 7316

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

Step	Hyp	Ref	Expression
1		n0i 4267	. 2 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → ¬ (𝐴𝐹𝐵) = ∅)
2		ovprc1.1	. . 3 ⊢ Rel dom 𝐹
3	2	ovprc 7313	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
4	1, 3	nsyl2 141	1 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  dom cdm 5589  Rel wrel 5594  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  smatrcl  31746