oppfval2 - Mathbox for Zhi Wang

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Theorem oppfval2 49860

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 17918	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		1st2nd 8035	. . . . 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 6885	. . 3 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
5		df-ov 7413	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	eqtr4di 2814	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
7		1st2ndbr 8038	. . . 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 49859	. . 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 2796	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 2141 ⟨cop 4594 class class class wbr 5108 Rel wrel 5666 ‘cfv 6536 (class class class)co 7410 1^st c1st 7983 2^nd c2nd 7984 tpos ctpos 8220 Func cfunc 17910 oppFunc coppf 49845

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 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732

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 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7985 df-2nd 7986 df-tpos 8221 df-map 8825 df-ixp 8895 df-func 17914 df-oppf 49846

This theorem is referenced by: oppf1 49862 oppf2 49863 2oppffunc 49869 cofuoppf 49873 fulloppf 49886 fthoppf 49887 natoppf2 49953 opf11 50126 opf12 50127 ranval3 50354 islmd 50388

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