oppfval2 - Mathbox for Zhi Wang

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Theorem oppfval2 49696

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 17867	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		1st2nd 8005	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
3	1, 2	mpan 698	. . . 4 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	3	fveq2d 6856	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
5		df-ov 7384	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	eqtr4di 2805	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
7		1st2ndbr 8008	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	1, 7	mpan 698	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
9		oppfval 49695	. . 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 2787	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 1550 ∈ wcel 2132 ⟨cop 4578 class class class wbr 5090 Rel wrel 5641 ‘cfv 6506 (class class class)co 7381 1^st c1st 7953 2^nd c2nd 7954 tpos ctpos 8189 Func cfunc 17859 oppFunc coppf 49681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-1st 7955 df-2nd 7956 df-tpos 8190 df-map 8794 df-ixp 8865 df-func 17863 df-oppf 49682

This theorem is referenced by: oppf1 49698 oppf2 49699 2oppffunc 49705 cofuoppf 49709 fulloppf 49722 fthoppf 49723 natoppf2 49789 opf11 49962 opf12 49963 ranval3 50190 islmd 50224

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