idfu1a - Mathbox for Zhi Wang

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Theorem idfu1a 50421

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 2763	. 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 50417	. . 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 50419	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	7, 9	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	1, 2, 6, 10	idfu1 18081	1 ⊢ (𝜑 → ((1^st ‘𝐼)‘𝑋) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ‘cfv 6555 (class class class)co 7440 1^st c1st 8020 Basecbs 17413 Catccat 17864 Func cfunc 18055 id_funccidfu 18056

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 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 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-map 8874 df-ixp 8951 df-cat 17868 df-cid 17869 df-homf 17870 df-func 18059 df-idfu 18060

This theorem is used by: (None)

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