Theorem oppfrcl 49253

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

Step	Hyp	Ref	Expression
1		oppfrcl.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
2		oppfrcl.2	. . . 4 ⊢ Rel 𝑅
3	1, 2	oppfrcllem 49252	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
5		ndmfv 6860	. . . . 5 ⊢ (¬ 𝐹 ∈ dom oppFunc → ( oppFunc ‘𝐹) = ∅)
6	4, 5	eqtrid 2780	. . . 4 ⊢ (¬ 𝐹 ∈ dom oppFunc → 𝐺 = ∅)
7	6	necon1ai 2956	. . 3 ⊢ (𝐺 ≠ ∅ → 𝐹 ∈ dom oppFunc )
8	3, 7	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ dom oppFunc )
9		oppffn 49249	. . 3 ⊢ oppFunc Fn (V × V)
10	9	fndmi 6590	. 2 ⊢ dom oppFunc = (V × V)
11	8, 10	eleqtrdi 2843	1 ⊢ (𝜑 → 𝐹 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437  ∅c0 4282   × cxp 5617  dom cdm 5619  Rel wrel 5624  ‘cfv 6486   oppFunc coppf 49247

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 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-oppf 49248

This theorem is referenced by:  oppfrcl2  49254  2oppf  49257  funcoppc5  49270