idfu1sta - Mathbox for Zhi Wang

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Theorem idfu1sta 49226

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 2733	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2838	. . . 4 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49223	. . . 4 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17788	. 2 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 49225	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	9	reseq2d 5932	. 2 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝐶)))
11	7, 10	eqtr4d 2771	1 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 I cid 5513 ↾ cres 5621 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 Basecbs 17122 Catccat 17572 Func cfunc 17763 id_funccidfu 17764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 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 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-map 8758 df-ixp 8828 df-cat 17576 df-cid 17577 df-homf 17578 df-func 17767 df-idfu 17768

This theorem is referenced by: imaidfu2lem 49234 imaidfu 49235 imaidfu2 49236 idsubc 49285

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