Theorem ovrcl 7437

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 4292	. 2 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → ¬ (𝐴𝐹𝐵) = ∅)
2		ovprc1.1	. . 3 ⊢ Rel dom 𝐹
3	2	ovprc 7434	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅)
4	1, 3	nsyl2 141	1 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  dom cdm 5647  Rel wrel 5652  (class class class)co 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-dm 5657  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  ghmquskerco  19324  smatrcl  34093  grimprop  48505  grimuhgr  48509  grimcnv  48510  grimco  48511  uhgrimisgrgric  48553  grlimprop  48606  grlimprop2  48608