Theorem oppfrcl2 49374

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 49373	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (V × V))
6	1, 5	eqeltrrd 2837	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7		0nelxp 5658	. . 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 4852	. . 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 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ∅c0 4285  ⟨cop 4586   × cxp 5622  Rel wrel 5629  ‘cfv 6492   oppFunc coppf 49367

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-oppf 49368

This theorem is referenced by:  oppfrcl3  49375  oppf1st2nd  49376