Theorem oppfrcl2 49025

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 49024	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (V × V))
6	1, 5	eqeltrrd 2835	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		0nelxp 5688	. . 3 ⊢ ¬ ∅ ∈ (V × V)
8		nelne2 3030	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ∅ ∈ (V × V)) → ⟨𝐴, 𝐵⟩ ≠ ∅)
9	6, 7, 8	sylancl 586	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ≠ ∅)
10		opprc 4872	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
11	10	necon1ai 2959	. 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 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308  ⟨cop 4607   × cxp 5652  Rel wrel 5659  ‘cfv 6530  oppFunccoppf 49019

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-oppf 49020

This theorem is referenced by:  oppfrcl3  49026  oppf1st2nd  49027