Theorem oppfrcl 49110

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 49109	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
5		ndmfv 6875	. . . . 5 ⊢ (¬ 𝐹 ∈ dom oppFunc → ( oppFunc ‘𝐹) = ∅)
6	4, 5	eqtrid 2776	. . . 4 ⊢ (¬ 𝐹 ∈ dom oppFunc → 𝐺 = ∅)
7	6	necon1ai 2952	. . 3 ⊢ (𝐺 ≠ ∅ → 𝐹 ∈ dom oppFunc )
8	3, 7	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ dom oppFunc )
9		oppffn 49106	. . 3 ⊢ oppFunc Fn (V × V)
10	9	fndmi 6604	. 2 ⊢ dom oppFunc = (V × V)
11	8, 10	eleqtrdi 2838	1 ⊢ (𝜑 → 𝐹 ∈ (V × V))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3444  ∅c0 4292   × cxp 5629  dom cdm 5631  Rel wrel 5636  ‘cfv 6499   oppFunc coppf 49104

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-oppf 49105

This theorem is referenced by:  oppfrcl2  49111  2oppf  49114  funcoppc5  49127