opf11 - Mathbox for Zhi Wang

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Theorem opf11 49893

Description: The object part of the op functor on functor categories. Lemma for fucoppc 49900. (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 6829	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋))
3		opf11.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))
4	3	fvresd 6847	. . 3 ⊢ (𝜑 → (( oppFunc ↾ (𝐶 Func 𝐷))‘𝑋) = ( oppFunc ‘𝑋))
5		oppfval2 49627	. . . 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 2778	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘𝑋), tpos (2^nd ‘𝑋)⟩)
8		fvex 6840	. . 3 ⊢ (1^st ‘𝑋) ∈ V
9		fvex 6840	. . . 4 ⊢ (2^nd ‘𝑋) ∈ V
10	9	tposex 8200	. . 3 ⊢ tpos (2^nd ‘𝑋) ∈ V
11	8, 10	op1std 7941	. 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 1547 ∈ wcel 2119 ⟨cop 4561 ↾ cres 5620 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 tpos ctpos 8165 Func cfunc 17812 oppFunc coppf 49612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-tpos 8166 df-map 8765 df-ixp 8836 df-func 17816 df-oppf 49613

This theorem is referenced by: fucoppcid 49898 fucoppcco 49899 oppfdiag1 49904

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