2oppf - Mathbox for Zhi Wang

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Theorem 2oppf 49619

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 6847	. . 3 ⊢ (1^st ‘𝐹) ∈ V
2		fvex 6847	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
3	2	tposex 8203	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
4		oppfvalg 49613	. . 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 693	. 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 7363	. . 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 49615	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (V × V))
11		1st2nd2 7974	. . . . . . 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 49618	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1^st ‘𝐺) = (1^st ‘𝐹) ∧ (2^nd ‘𝐺) = tpos (2^nd ‘𝐹))))
14		eqopi 7971	. . . . 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 6838	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ( oppFunc ‘⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
17	6, 16	eqtr4id 2791	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = ( oppFunc ‘𝐺))
18	7, 8, 9, 12	oppfrcl3 49617	. . . . 5 ⊢ (𝜑 → (Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)))
19		tpostpos2 8190	. . . . 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 4824	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩ = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
22		0nelrel0 5684	. . . . . . 7 ⊢ (Rel dom (2^nd ‘𝐹) → ¬ ∅ ∈ dom (2^nd ‘𝐹))
23	18, 22	simpl2im 503	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ dom (2^nd ‘𝐹))
24		reldmtpos 8177	. . . . . 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 8174	. . . . 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 4475	. . 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 2782	. 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 2796	1 ⊢ (𝜑 → ( oppFunc ‘𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ∅c0 4274 ifcif 4467 ⟨cop 4574 × cxp 5622 dom cdm 5624 Rel wrel 5629 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 tpos ctpos 8168 oppFunc coppf 49609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-tpos 8169 df-oppf 49610

This theorem is referenced by: oppff1 49635 oppff1o 49636 natoppfb 49718 cmddu 50155

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