Theorem oppfrcl 49934

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 49933	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		oppfrcl.3	. . . . 5 ⊢ 𝐺 = ( oppFunc ‘𝐹)
5		ndmfv 6913	. . . . 5 ⊢ (¬ 𝐹 ∈ dom oppFunc → ( oppFunc ‘𝐹) = ∅)
6	4, 5	eqtrid 2810	. . . 4 ⊢ (¬ 𝐹 ∈ dom oppFunc → 𝐺 = ∅)
7	6	necon1ai 2985	. . 3 ⊢ (𝐺 ≠ ∅ → 𝐹 ∈ dom oppFunc )
8	3, 7	syl 18	. 2 ⊢ (𝜑 → 𝐹 ∈ dom oppFunc )
9		oppffn 49930	. . 3 ⊢ oppFunc Fn (V × V)
10	9	fndmi 6639	. 2 ⊢ dom oppFunc = (V × V)
11	8, 10	eleqtrdi 2873	1 ⊢ (𝜑 → 𝐹 ∈ (V × V))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286   × cxp 5659  dom cdm 5661  Rel wrel 5666  ‘cfv 6536   oppFunc coppf 49928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-oppf 49929

This theorem is used by:  oppfrcl2  49935  2oppf  49938  funcoppc5  49951