Theorem oppfrcl2 49124

Description: If an opposite functor of a class is a functor, then the two components of the original class must be sets. (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
oppfrcl.1	⊢ (𝜑 → 𝐺 ∈ 𝑅)
oppfrcl.2	⊢ Rel 𝑅
oppfrcl.3	⊢ 𝐺 = ( oppFunc ‘𝐹)
oppfrcl2.4	⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)

Assertion

Ref	Expression
oppfrcl2	⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem oppfrcl2

Step	Hyp	Ref	Expression
1		oppfrcl2.4	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)
2		oppfrcl.1	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
3		oppfrcl.2	. . . . 5 ⊢ Rel 𝑅
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
5	2, 3, 4	oppfrcl 49123	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (V × V))
6	1, 5	eqeltrrd 2829	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		0nelxp 5653	. . 3 ⊢ ¬ ∅ ∈ (V × V)
8		nelne2 3023	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → ⟨𝐴, 𝐵⟩ ≠ ∅)
9	6, 7, 8	sylancl 586	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ≠ ∅)
10		opprc 4847	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
11	10	necon1ai 2952	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
12	9, 11	syl 17	1 ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436  ∅c0 4284  ⟨cop 4583   × cxp 5617  Rel wrel 5624  ‘cfv 6482   oppFunc coppf 49117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-oppf 49118

This theorem is referenced by:  oppfrcl3  49125  oppf1st2nd  49126