idfu1sta - Mathbox for Zhi Wang

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Theorem idfu1sta 49576

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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2841	. . . 4 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49573	. . . 4 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17846	. 2 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 49575	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	9	reseq2d 5944	. 2 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝐶)))
11	7, 10	eqtr4d 2774	1 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 I cid 5525 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 Basecbs 17179 Catccat 17630 Func cfunc 17821 id_funccidfu 17822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-map 8775 df-ixp 8846 df-cat 17634 df-cid 17635 df-homf 17636 df-func 17825 df-idfu 17826

This theorem is referenced by: imaidfu2lem 49584 imaidfu 49585 imaidfu2 49586 idsubc 49635

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