idfu1a - Mathbox for Zhi Wang

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Theorem idfu1a 49009

Description: Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)

**Hypotheses**
Ref	Expression
idfu2nda.i	⊢ 𝐼 = (id_func‘𝐶)
idfu2nda.d	⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
idfu2nda.b	⊢ (𝜑 → 𝐵 = (Base‘𝐷))
idfu2nda.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
idfu1a	⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)

**Proof of Theorem idfu1a**
Step	Hyp	Ref	Expression
1		idfu2nda.i	. 2 ⊢ 𝐼 = (id_func‘𝐶)
2		eqid 2735	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2839	. . 3 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49006	. . 3 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		idfu2nda.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 49007	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	7, 9	eleqtrd 2836	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	1, 2, 6, 10	idfu1 17891	1 ⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6530 (class class class)co 7403 1^st c1st 7984 Basecbs 17226 Catccat 17674 Func cfunc 17865 id_funccidfu 17866

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7986 df-2nd 7987 df-map 8840 df-ixp 8910 df-cat 17678 df-cid 17679 df-homf 17680 df-func 17869 df-idfu 17870

This theorem is referenced by: (None)

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