Theorem 2oppf 49028

Description: The double opposite functor is the original functor. (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
2oppf	⊢ (𝜑 → (oppFunc‘𝐺) = 𝐹)

Proof of Theorem 2oppf

Step	Hyp	Ref	Expression
1		fvex 6888	. . 3 ⊢ (1st ‘𝐹) ∈ V
2		fvex 6888	. . . 4 ⊢ (2nd ‘𝐹) ∈ V
3	2	tposex 8257	. . 3 ⊢ tpos (2nd ‘𝐹) ∈ V
4		oppfvalg 49022	. . 3 ⊢ (((1st ‘𝐹) ∈ V ∧ tpos (2nd ‘𝐹) ∈ V) → ((1st ‘𝐹)oppFunctpos (2nd ‘𝐹)) = if((Rel tpos (2nd ‘𝐹) ∧ Rel dom tpos (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩, ∅))
5	1, 3, 4	mp2an 692	. 2 ⊢ ((1st ‘𝐹)oppFunctpos (2nd ‘𝐹)) = if((Rel tpos (2nd ‘𝐹) ∧ Rel dom tpos (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩, ∅)
6		df-ov 7406	. . 3 ⊢ ((1st ‘𝐹)oppFunctpos (2nd ‘𝐹)) = (oppFunc‘⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
7		oppfrcl.1	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑅)
8		oppfrcl.2	. . . . . 6 ⊢ Rel 𝑅
9		oppfrcl.3	. . . . . 6 ⊢ 𝐺 = (oppFunc‘𝐹)
10	7, 8, 9	oppfrcl 49024	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (V × V))
11		1st2nd2 8025	. . . . . . 7 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
13	7, 8, 9, 12	oppf1st2nd 49027	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = (1st ‘𝐹) ∧ (2nd ‘𝐺) = tpos (2nd ‘𝐹))))
14		eqopi 8022	. . . . 5 ⊢ ((𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = (1st ‘𝐹) ∧ (2nd ‘𝐺) = tpos (2nd ‘𝐹))) → 𝐺 = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
16	15	fveq2d 6879	. . 3 ⊢ (𝜑 → (oppFunc‘𝐺) = (oppFunc‘⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩))
17	6, 16	eqtr4id 2789	. 2 ⊢ (𝜑 → ((1st ‘𝐹)oppFunctpos (2nd ‘𝐹)) = (oppFunc‘𝐺))
18	7, 8, 9, 12	oppfrcl3 49026	. . . . 5 ⊢ (𝜑 → (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)))
19		tpostpos2 8244	. . . . 5 ⊢ ((Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹)) → tpos tpos (2nd ‘𝐹) = (2nd ‘𝐹))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → tpos tpos (2nd ‘𝐹) = (2nd ‘𝐹))
21	20	opeq2d 4856	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
22		0nelrel0 5714	. . . . . . 7 ⊢ (Rel dom (2nd ‘𝐹) → ¬ ∅ ∈ dom (2nd ‘𝐹))
23	18, 22	simpl2im 503	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ dom (2nd ‘𝐹))
24		reldmtpos 8231	. . . . . 6 ⊢ (Rel dom tpos (2nd ‘𝐹) ↔ ¬ ∅ ∈ dom (2nd ‘𝐹))
25	23, 24	sylibr 234	. . . . 5 ⊢ (𝜑 → Rel dom tpos (2nd ‘𝐹))
26		reltpos 8228	. . . . 5 ⊢ Rel tpos (2nd ‘𝐹)
27	25, 26	jctil 519	. . . 4 ⊢ (𝜑 → (Rel tpos (2nd ‘𝐹) ∧ Rel dom tpos (2nd ‘𝐹)))
28	27	iftrued 4508	. . 3 ⊢ (𝜑 → if((Rel tpos (2nd ‘𝐹) ∧ Rel dom tpos (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩, ∅) = ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩)
29	21, 28, 12	3eqtr4d 2780	. 2 ⊢ (𝜑 → if((Rel tpos (2nd ‘𝐹) ∧ Rel dom tpos (2nd ‘𝐹)), ⟨(1st ‘𝐹), tpos tpos (2nd ‘𝐹)⟩, ∅) = 𝐹)
30	5, 17, 29	3eqtr3a 2794	1 ⊢ (𝜑 → (oppFunc‘𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ifcif 4500  ⟨cop 4607   × cxp 5652  dom cdm 5654  Rel wrel 5659  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  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-pow 5335  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-pw 4577  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-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-tpos 8223  df-oppf 49020

This theorem is referenced by:  2oppffunc  49037