idfu1a - Mathbox for Zhi Wang

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Theorem idfu1a 50093

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 2760	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2865	. . 3 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 50089	. . 3 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		idfu2nda.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 50091	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	7, 9	eleqtrd 2862	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	1, 2, 6, 10	idfu1 17994	1 ⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ‘cfv 6535 (class class class)co 7416 1^st c1st 7990 Basecbs 17326 Catccat 17777 Func cfunc 17968 id_funccidfu 17969

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7992 df-2nd 7993 df-map 8835 df-ixp 8912 df-cat 17781 df-cid 17782 df-homf 17783 df-func 17972 df-idfu 17973

This theorem is used by: (None)

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