oppf1 - Mathbox for Zhi Wang

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Theorem oppf1 49917

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 49915	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
3		fvex 6894	. . 3 ⊢ (1^st ‘𝐹) ∈ V
4		fvex 6894	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
5	4	tposex 8252	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
6	3, 5	op1std 7992	. 2 ⊢ (( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩ → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))
7	1, 2, 6	3syl 19	1 ⊢ (𝜑 → (1^st ‘( oppFunc ‘𝐹)) = (1^st ‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 ⟨cop 4595 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 tpos ctpos 8217 Func cfunc 17906 oppFunc coppf 49900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 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 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 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 7982 df-2nd 7983 df-tpos 8218 df-map 8822 df-ixp 8892 df-func 17910 df-oppf 49901

This theorem is referenced by: oppfdiag1 50192 oppfdiag 50194 lmddu 50445 lmdran 50449

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