Theorem oppfrcl2 49229

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 49228	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (V × V))
6	1, 5	eqeltrrd 2832	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		0nelxp 5648	. . 3 ⊢ ¬ ∅ ∈ (V × V)
8		nelne2 3026	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → ⟨𝐴, 𝐵⟩ ≠ ∅)
9	6, 7, 8	sylancl 586	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ≠ ∅)
10		opprc 4845	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
11	10	necon1ai 2955	. 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 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4280  ⟨cop 4579   × cxp 5612  Rel wrel 5619  ‘cfv 6481   oppFunc coppf 49222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-oppf 49223

This theorem is referenced by:  oppfrcl3  49230  oppf1st2nd  49231