idfu1a - Mathbox for Zhi Wang

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Theorem idfu1a 49728

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 2764	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2869	. . 3 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49724	. . 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 49726	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	7, 9	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	1, 2, 6, 10	idfu1 17915	1 ⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1562 ∈ wcel 2144 ‘cfv 6523 (class class class)co 7398 1^st c1st 7970 Basecbs 17247 Catccat 17698 Func cfunc 17889 id_funccidfu 17890

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-1st 7972 df-2nd 7973 df-map 8812 df-ixp 8882 df-cat 17702 df-cid 17703 df-homf 17704 df-func 17893 df-idfu 17894

This theorem is referenced by: (None)

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