Theorem oppfrcl 49626

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 49625	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
5		ndmfv 6860	. . . . 5 ⊢ (¬ 𝐹 ∈ dom oppFunc → ( oppFunc ‘𝐹) = ∅)
6	4, 5	eqtrid 2786	. . . 4 ⊢ (¬ 𝐹 ∈ dom oppFunc → 𝐺 = ∅)
7	6	necon1ai 2961	. . 3 ⊢ (𝐺 ≠ ∅ → 𝐹 ∈ dom oppFunc )
8	3, 7	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ dom oppFunc )
9		oppffn 49622	. . 3 ⊢ oppFunc Fn (V × V)
10	9	fndmi 6590	. 2 ⊢ dom oppFunc = (V × V)
11	8, 10	eleqtrdi 2849	1 ⊢ (𝜑 → 𝐹 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  ∅c0 4262   × cxp 5617  dom cdm 5619  Rel wrel 5624  ‘cfv 6486   oppFunc coppf 49620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  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 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-oppf 49621

This theorem is referenced by:  oppfrcl2  49627  2oppf  49630  funcoppc5  49643