2oppf - Mathbox for Zhi Wang

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Theorem 2oppf 49257

Description: The double opposite functor is the original functor. Remark 3.42 of [Adamek] p. 39. (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 6841	. . 3 ⊢ (1^st ‘𝐹) ∈ V
2		fvex 6841	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
3	2	tposex 8196	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
4		oppfvalg 49251	. . 3 ⊢ (((1^st ‘𝐹) ∈ V ∧ tpos (2^nd ‘𝐹) ∈ V) → ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = if((Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)), ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩, ∅))
5	1, 3, 4	mp2an 692	. 2 ⊢ ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = if((Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)), ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩, ∅)
6		df-ov 7355	. . 3 ⊢ ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), tpos (2^nd ‘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 49253	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (V × V))
11		1st2nd2 7966	. . . . . . 7 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
13	7, 8, 9, 12	oppf1st2nd 49256	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1^st ‘𝐺) = (1^st ‘𝐹) ∧ (2^nd ‘𝐺) = tpos (2^nd ‘𝐹))))
14		eqopi 7963	. . . . 5 ⊢ ((𝐺 ∈ (V × V) ∧ ((1^st ‘𝐺) = (1^st ‘𝐹) ∧ (2^nd ‘𝐺) = tpos (2^nd ‘𝐹))) → 𝐺 = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
16	15	fveq2d 6832	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ( oppFunc ‘⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
17	6, 16	eqtr4id 2787	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = ( oppFunc ‘𝐺))
18	7, 8, 9, 12	oppfrcl3 49255	. . . . 5 ⊢ (𝜑 → (Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)))
19		tpostpos2 8183	. . . . 5 ⊢ ((Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)) → tpos tpos (2^nd ‘𝐹) = (2^nd ‘𝐹))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → tpos tpos (2^nd ‘𝐹) = (2^nd ‘𝐹))
21	20	opeq2d 4831	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩ = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
22		0nelrel0 5679	. . . . . . 7 ⊢ (Rel dom (2^nd ‘𝐹) → ¬ ∅ ∈ dom (2^nd ‘𝐹))
23	18, 22	simpl2im 503	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ dom (2^nd ‘𝐹))
24		reldmtpos 8170	. . . . . 6 ⊢ (Rel dom tpos (2^nd ‘𝐹) ↔ ¬ ∅ ∈ dom (2^nd ‘𝐹))
25	23, 24	sylibr 234	. . . . 5 ⊢ (𝜑 → Rel dom tpos (2^nd ‘𝐹))
26		reltpos 8167	. . . . 5 ⊢ Rel tpos (2^nd ‘𝐹)
27	25, 26	jctil 519	. . . 4 ⊢ (𝜑 → (Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)))
28	27	iftrued 4482	. . 3 ⊢ (𝜑 → if((Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)), ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩, ∅) = ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩)
29	21, 28, 12	3eqtr4d 2778	. 2 ⊢ (𝜑 → if((Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)), ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩, ∅) = 𝐹)
30	5, 17, 29	3eqtr3a 2792	1 ⊢ (𝜑 → ( oppFunc ‘𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ∅c0 4282 ifcif 4474 ⟨cop 4581 × cxp 5617 dom cdm 5619 Rel wrel 5624 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 tpos ctpos 8161 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-pow 5305 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-pw 4551 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-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-tpos 8162 df-oppf 49248

This theorem is referenced by: oppff1 49273 oppff1o 49274 natoppfb 49356 cmddu 49793

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