oppfval2 - Mathbox for Zhi Wang

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Theorem oppfval2 50458

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 18066	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		1st2nd 8074	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
3	1, 2	mpan 703	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	3	fveq2d 6912	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
5		df-ov 7446	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	eqtr4di 2816	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
7		1st2ndbr 8077	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	1, 7	mpan 703	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9		oppfval 50457	. . 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 2798	1 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 class class class wbr 5106 Rel wrel 5660 ‘cfv 6558 (class class class)co 7443 1^st c1st 8023 2^nd c2nd 8024 tpos ctpos 8270 Func cfunc 18058 oppFunc coppf 50443

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-fv 6566 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-tpos 8271 df-map 8877 df-ixp 8954 df-func 18062 df-oppf 50444

This theorem is used by: oppf1 50460 oppf2 50461 2oppffunc 50467 cofuoppf 50471 fulloppf 50484 fthoppf 50485 natoppf2 50557 opf11 50743 opf12 50744 ranval3 51019 islmd 51053

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