oppfval2 - Mathbox for Zhi Wang

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Theorem oppfval2 49864

Description: Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)

**Assertion**
Ref	Expression
oppfval2	⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)

**Proof of Theorem oppfval2**
Step	Hyp	Ref	Expression
1		relfunc 17922	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		1st2nd 8039	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
3	1, 2	mpan 702	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	3	fveq2d 6889	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
5		df-ov 7417	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	eqtr4di 2823	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
7		1st2ndbr 8042	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	1, 7	mpan 702	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9		oppfval 49863	. . 3 ⊢ ((1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹) → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
10	8, 9	syl 18	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
11	6, 10	eqtrd 2805	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1568 ∈ wcel 2150 ⟨cop 4600 class class class wbr 5114 Rel wrel 5670 ‘cfv 6540 (class class class)co 7414 1^st c1st 7987 2^nd c2nd 7988 tpos ctpos 8224 Func cfunc 17914 oppFunc coppf 49849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-ral 3087 df-rex 3097 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 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 5560 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-tpos 8225 df-map 8829 df-ixp 8899 df-func 17918 df-oppf 49850

This theorem is referenced by: oppf1 49866 oppf2 49867 2oppffunc 49873 cofuoppf 49877 fulloppf 49890 fthoppf 49891 natoppf2 49957 opf11 50130 opf12 50131 ranval3 50358 islmd 50392

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