oppf1 - Mathbox for Zhi Wang

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Theorem oppf1 50586

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 50584	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
3		fvex 6929	. . 3 ⊢ (1^st ‘𝐹) ∈ V
4		fvex 6929	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
5	4	tposex 8315	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
6	3, 5	op1std 8045	. 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⟨cop 4840 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 tpos ctpos 8280 Func cfunc 18079 oppFunc coppf 50569

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 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

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 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-tpos 8281 df-map 8887 df-ixp 8964 df-func 18083 df-oppf 50570

This theorem is used by: oppfdiag1 50880 oppfdiag 50882 lmddu 51181 lmdran 51185

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