oppfval2 - Mathbox for Zhi Wang

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Theorem oppfval2 49600

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 17818	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		1st2nd 7981	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
3	1, 2	mpan 691	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	3	fveq2d 6833	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
5		df-ov 7359	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	eqtr4di 2788	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
7		1st2ndbr 7984	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	1, 7	mpan 691	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9		oppfval 49599	. . 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 17	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
11	6, 10	eqtrd 2770	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 1542 ∈ wcel 2114 ⟨cop 4563 class class class wbr 5074 Rel wrel 5625 ‘cfv 6487 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 tpos ctpos 8164 Func cfunc 17810 oppFunc coppf 49585

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-fv 6495 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-tpos 8165 df-map 8764 df-ixp 8835 df-func 17814 df-oppf 49586

This theorem is referenced by: oppf1 49602 oppf2 49603 2oppffunc 49609 cofuoppf 49613 fulloppf 49626 fthoppf 49627 natoppf2 49693 opf11 49866 opf12 49867 ranval3 50094 islmd 50128

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