opf11 - Mathbox for Zhi Wang

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Theorem opf11 50480

Description: The object part of the op functor on functor categories. Lemma for fucoppc 50487. (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 6885	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6903	. . 3 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 50214	. . . 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 2800	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
8		fvex 6896	. . 3 ⊢ (1^st ‘𝑋) ∈ V
9		fvex 6896	. . . 4 ⊢ (2^nd ‘𝑋) ∈ V
10	9	tposex 8270	. . 3 ⊢ tpos (2^nd ‘𝑋) ∈ V
11	8, 10	op1std 8009	. 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 ↾ cres 5653 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 tpos ctpos 8235 Func cfunc 18022 oppFunc coppf 50199

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7421 df-oprab 7422 df-mpo 7423 df-1st 7999 df-2nd 8000 df-tpos 8236 df-map 8842 df-ixp 8919 df-func 18026 df-oppf 50200

This theorem is used by: fucoppcid 50485 fucoppcco 50486 oppfdiag1 50491

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