2oppf - Mathbox for Zhi Wang

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Theorem 2oppf 49717

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 6876	. . 3 ⊢ (1^st ‘𝐹) ∈ V
2		fvex 6876	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
3	2	tposex 8235	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
4		oppfvalg 49711	. . 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 702	. 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 7395	. . 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 49713	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (V × V))
11		1st2nd2 8005	. . . . . . 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 49716	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1^st ‘𝐺) = (1^st ‘𝐹) ∧ (2^nd ‘𝐺) = tpos (2^nd ‘𝐹))))
14		eqopi 8002	. . . . 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 6867	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐺) = ( oppFunc ‘⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩))
17	6, 16	eqtr4id 2815	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc tpos (2^nd ‘𝐹)) = ( oppFunc ‘𝐺))
18	7, 8, 9, 12	oppfrcl3 49715	. . . . 5 ⊢ (𝜑 → (Rel (2^nd ‘𝐹) ∧ Rel dom (2^nd ‘𝐹)))
19		tpostpos2 8222	. . . . 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 4837	. . 3 ⊢ (𝜑 → ⟨(1^st ‘𝐹), tpos tpos (2^nd ‘𝐹)⟩ = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
22		0nelrel0 5705	. . . . . . 7 ⊢ (Rel dom (2^nd ‘𝐹) → ¬ ∅ ∈ dom (2^nd ‘𝐹))
23	18, 22	simpl2im 511	. . . . . 6 ⊢ (𝜑 → ¬ ∅ ∈ dom (2^nd ‘𝐹))
24		reldmtpos 8209	. . . . . 6 ⊢ (Rel dom tpos (2^nd ‘𝐹) ↔ ¬ ∅ ∈ dom (2^nd ‘𝐹))
25	23, 24	sylibr 236	. . . . 5 ⊢ (𝜑 → Rel dom tpos (2^nd ‘𝐹))
26		reltpos 8206	. . . . 5 ⊢ Rel tpos (2^nd ‘𝐹)
27	25, 26	jctil 527	. . . 4 ⊢ (𝜑 → (Rel tpos (2^nd ‘𝐹) ∧ Rel dom tpos (2^nd ‘𝐹)))
28	27	iftrued 4487	. . 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 2806	. 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 2820	1 ⊢ (𝜑 → ( oppFunc ‘𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ∅c0 4285 ifcif 4479 ⟨cop 4587 × cxp 5643 dom cdm 5645 Rel wrel 5650 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 2^nd c2nd 7965 tpos ctpos 8200 oppFunc coppf 49707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-tpos 8201 df-oppf 49708

This theorem is referenced by: oppff1 49733 oppff1o 49734 natoppfb 49816 cmddu 50253

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