idfu1sta - Mathbox for Zhi Wang

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Theorem idfu1sta 49096

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))

**Assertion**
Ref	Expression
idfu1sta	⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

**Proof of Theorem idfu1sta**
Step	Hyp	Ref	Expression
1		idfu2nda.i	. . 3 ⊢ 𝐼 = (id_func‘𝐶)
2		eqid 2729	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2833	. . . 4 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49093	. . . 4 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17786	. 2 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 49095	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	9	reseq2d 5930	. 2 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝐶)))
11	7, 10	eqtr4d 2767	1 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 I cid 5513 ↾ cres 5621 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 Basecbs 17120 Catccat 17570 Func cfunc 17761 id_funccidfu 17762

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-map 8755 df-ixp 8825 df-cat 17574 df-cid 17575 df-homf 17576 df-func 17765 df-idfu 17766

This theorem is referenced by: imaidfu2lem 49104 imaidfu 49105 imaidfu2 49106 idsubc 49155

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