Theorem aovrcl 48181

Description: Reverse closure for an operation value, analogous to afvvv 48137. In contrast to ovrcl 7449, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aovprc.1	⊢ Rel dom 𝐹

Assertion

Ref	Expression
aovrcl	⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem aovrcl

Step	Hyp	Ref	Expression
1		df-aov 48113	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	eleq1i 2851	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3		afvvdm 48133	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
4		df-br 5103	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
5		aovprc.1	. . . . 5 ⊢ Rel dom 𝐹
6	5	brrelex12i 5702	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	4, 6	sylbir 238	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	3, 7	syl 18	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	2, 8	sylbi 220	1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  Vcvv 3450  ⟨cop 4589   class class class wbr 5102  dom cdm 5647  Rel wrel 5652  '''cafv 48109   ((caov 48110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-iota 6483  df-fun 6529  df-fv 6535  df-aiota 48077  df-dfat 48111  df-afv 48112  df-aov 48113

This theorem is used by: (None)