opf11 - Mathbox for Zhi Wang

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Theorem opf11 49372

Description: The object part of the op functor on functor categories. Lemma for fucoppc 49379. (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 6862	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6880	. . 3 ⊢ (𝜑 → ((oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = (oppFunc‘𝑋))
5		oppfval2 49114	. . . 4 ⊢ (𝑋 ∈ (𝐶 Func 𝐷) → (oppFunc‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → (oppFunc‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
7	2, 4, 6	3eqtrd 2769	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
8		fvex 6873	. . 3 ⊢ (1^st ‘𝑋) ∈ V
9		fvex 6873	. . . 4 ⊢ (2^nd ‘𝑋) ∈ V
10	9	tposex 8241	. . 3 ⊢ tpos (2^nd ‘𝑋) ∈ V
11	8, 10	op1std 7980	. 2 ⊢ ((𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩ → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))
12	7, 11	syl 17	1 ⊢ (𝜑 → (1^st ‘(𝐹‘𝑋)) = (1^st ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⟨cop 4597 ↾ cres 5642 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 tpos ctpos 8206 Func cfunc 17822 oppFunccoppf 49099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-1st 7970 df-2nd 7971 df-tpos 8207 df-map 8803 df-ixp 8873 df-func 17826 df-oppf 49100

This theorem is referenced by: fucoppcid 49377 fucoppcco 49378 oppfdiag1 49383

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