Theorem ovrcl 7443

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 4326	. 2 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → ¬ (𝐴𝐹𝐵) = ∅)
2		ovprc1.1	. . 3 ⊢ Rel dom 𝐹
3	2	ovprc 7440	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
4	1, 3	nsyl2 141	1 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ∅c0 4315  dom cdm 5667  Rel wrel 5672  (class class class)co 7402

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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-dm 5677  df-iota 6486  df-fv 6542  df-ov 7405

This theorem is referenced by:  ghmquskerco  33024  smatrcl  33295