oppf1 - Mathbox for Zhi Wang

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Theorem oppf1 49171

Description: Value of the object part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)

**Hypothesis**
Ref	Expression
oppf1.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
oppf1	⊢ (𝜑 → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))

**Proof of Theorem oppf1**
Step	Hyp	Ref	Expression
1		oppf1.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
2		oppfval2 49169	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
3		fvex 6830	. . 3 ⊢ (1^st ‘𝐹) ∈ V
4		fvex 6830	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
5	4	tposex 8185	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
6	3, 5	op1std 7926	. 2 ⊢ (( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩ → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))
7	1, 2, 6	3syl 18	1 ⊢ (𝜑 → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ⟨cop 4577 ‘cfv 6476 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 tpos ctpos 8150 Func cfunc 17756 oppFunc coppf 49154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917 df-tpos 8151 df-map 8747 df-ixp 8817 df-func 17760 df-oppf 49155

This theorem is referenced by: oppfdiag1 49446 oppfdiag 49448 lmddu 49699 lmdran 49703

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