opf11 - Mathbox for Zhi Wang

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Theorem opf11 50061

Description: The object part of the op functor on functor categories. Lemma for fucoppc 50068. (Contributed by Zhi Wang, 18-Nov-2025.)

**Hypotheses**
Ref	Expression
opf11.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
opf11.x	⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
opf11	⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))

**Proof of Theorem opf11**
Step	Hyp	Ref	Expression
1		opf11.f	. . . 4 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))
2	1	fveq1d 6881	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6899	. . 3 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 49795	. . . 4 ⊢ (𝑋 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
7	2, 4, 6	3eqtrd 2808	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
8		fvex 6892	. . 3 ⊢ (1^st ‘𝑋) ∈ V
9		fvex 6892	. . . 4 ⊢ (2^nd ‘𝑋) ∈ V
10	9	tposex 8252	. . 3 ⊢ tpos (2^nd ‘𝑋) ∈ V
11	8, 10	op1std 7992	. 2 ⊢ ((𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩ → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
12	7, 11	syl 18	1 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 ⟨cop 4597 ↾ cres 5661 ‘cfv 6534 (class class class)co 7408 1^st c1st 7980 2^nd c2nd 7981 tpos ctpos 8217 Func cfunc 17907 oppFunc coppf 49780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7982 df-2nd 7983 df-tpos 8218 df-map 8822 df-ixp 8892 df-func 17911 df-oppf 49781

This theorem is referenced by: fucoppcid 50066 fucoppcco 50067 oppfdiag1 50072

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