Theorem oppfrcl 49024

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

Hypotheses

Ref	Expression
oppfrcl.1	⊢ (𝜑 → 𝐺 ∈ 𝑅)
oppfrcl.2	⊢ Rel 𝑅
oppfrcl.3	⊢ 𝐺 = (oppFunc‘𝐹)

Assertion

Ref	Expression
oppfrcl	⊢ (𝜑 → 𝐹 ∈ (V × V))

Proof of Theorem oppfrcl

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppfrcl.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
2		oppfrcl.2	. . . 4 ⊢ Rel 𝑅
3	1, 2	oppfrcllem 49023	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = (oppFunc‘𝐹)
5		ndmfv 6910	. . . . 5 ⊢ (¬ 𝐹 ∈ dom oppFunc → (oppFunc‘𝐹) = ∅)
6	4, 5	eqtrid 2782	. . . 4 ⊢ (¬ 𝐹 ∈ dom oppFunc → 𝐺 = ∅)
7	6	necon1ai 2959	. . 3 ⊢ (𝐺 ≠ ∅ → 𝐹 ∈ dom oppFunc)
8	3, 7	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ dom oppFunc)
9		df-oppf 49020	. . 3 ⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
10		opex 5439	. . . 4 ⊢ ⟨𝑓, tpos 𝑔⟩ ∈ V
11		0ex 5277	. . . 4 ⊢ ∅ ∈ V
12	10, 11	ifex 4551	. . 3 ⊢ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅) ∈ V
13	9, 12	dmmpo 8068	. 2 ⊢ dom oppFunc = (V × V)
14	8, 13	eleqtrdi 2844	1 ⊢ (𝜑 → 𝐹 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308  ifcif 4500  ⟨cop 4607   × cxp 5652  dom cdm 5654  Rel wrel 5659  ‘cfv 6530  tpos ctpos 8222  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:  oppfrcl2  49025  2oppf  49028  funcoppc5  49036