2oppf - Mathbox for Zhi Wang

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Theorem 2oppf 49951

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 6901	. . 3 ⊢ (1^st ‘𝐹) ∈ V
2		fvex 6901	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
3	2	tposex 8265	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
4		oppfvalg 49945	. . 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 705	. 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 7426	. . 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 49947	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (V × V))
11		1st2nd2 8034	. . . . . . 7 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
12	10, 11	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
13	7, 8, 9, 12	oppf1st2nd 49950	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1^st ‘𝐺) = (1^st ‘𝐹) ∧ (2^nd ‘𝐺) = tpos (2^nd ‘𝐹))))
14		eqopi 8031	. . . . 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 18	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
16	15	fveq2d 6892	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ( oppFunc ‘⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
17	6, 16	eqtr4id 2820	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = ( oppFunc ‘𝐺))
18	7, 8, 9, 12	oppfrcl3 49949	. . . . 5 ⊢ (𝜑 → (Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)))
19		tpostpos2 8252	. . . . 5 ⊢ ((Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)) → tpos tpos (2^nd ‘𝐹) = (2^nd ‘𝐹))
20	18, 19	syl 18	. . . 4 ⊢ (𝜑 → tpos tpos (2^nd ‘𝐹) = (2^nd ‘𝐹))
21	20	opeq2d 4850	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩ = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
22		0nelrel0 5726	. . . . . . 7 ⊢ (Rel dom (2^nd ‘𝐹) → ¬ ∅ ∈ dom (2^nd ‘𝐹))
23	18, 22	simpl2im 513	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ dom (2^nd ‘𝐹))
24		reldmtpos 8239	. . . . . 6 ⊢ (Rel dom tpos (2^nd ‘𝐹) ↔ ¬ ∅ ∈ dom (2^nd ‘𝐹))
25	23, 24	sylibr 237	. . . . 5 ⊢ (𝜑 → Rel dom tpos (2^nd ‘𝐹))
26		reltpos 8236	. . . . 5 ⊢ Rel tpos (2^nd ‘𝐹)
27	25, 26	jctil 529	. . . 4 ⊢ (𝜑 → (Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)))
28	27	iftrued 4500	. . 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 2811	. 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 2825	1 ⊢ (𝜑 → ( oppFunc ‘𝐺) = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ∅c0 4289 ifcif 4492 ⟨cop 4600 × cxp 5664 dom cdm 5666 Rel wrel 5671 ‘cfv 6543 (class class class)co 7423 1^st c1st 7993 2^nd c2nd 7994 tpos ctpos 8230 oppFunc coppf 49941

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 7995 df-2nd 7996 df-tpos 8231 df-oppf 49942

This theorem is used by: oppff1 49967 oppff1o 49968 natoppfb 50050 cmddu 50487

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