idfu1sta - Mathbox for Zhi Wang

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Theorem idfu1sta 49413

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 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2842	. . . 4 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 49410	. . . 4 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17807	. 2 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 49412	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	9	reseq2d 5939	. 2 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝐶)))
11	7, 10	eqtr4d 2775	1 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 I cid 5519 ↾ cres 5627 ‘cfv 6493 (class class class)co 7360 1^st c1st 7933 Basecbs 17140 Catccat 17591 Func cfunc 17782 id_funccidfu 17783

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 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-map 8769 df-ixp 8840 df-cat 17595 df-cid 17596 df-homf 17597 df-func 17786 df-idfu 17787

This theorem is referenced by: imaidfu2lem 49421 imaidfu 49422 imaidfu2 49423 idsubc 49472

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