oppf1 - Mathbox for Zhi Wang

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Theorem oppf1 50065

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 50063	. 2 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1^st ‘𝐹), tpos (2^nd ‘𝐹)⟩)
3		fvex 6891	. . 3 ⊢ (1^st ‘𝐹) ∈ V
4		fvex 6891	. . . 4 ⊢ (2^nd ‘𝐹) ∈ V
5	4	tposex 8258	. . 3 ⊢ tpos (2^nd ‘𝐹) ∈ V
6	3, 5	op1std 7996	. 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 2145 ⟨cop 4590 ‘cfv 6533 (class class class)co 7413 1^st c1st 7984 2^nd c2nd 7985 tpos ctpos 8223 Func cfunc 17943 oppFunc coppf 50048

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-tpos 8224 df-map 8828 df-ixp 8905 df-func 17947 df-oppf 50049

This theorem is used by: oppfdiag1 50340 oppfdiag 50342 lmddu 50593 lmdran 50597

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