idfu1sta - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > idfu1sta		Structured version Visualization version GIF version

Theorem idfu1sta 50012

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 2762	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		idfu2nda.d	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
4	1, 3	eqeltrrid 2867	. . . 4 ⊢ (𝜑 → (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
5		idfurcl 50009	. . . 4 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	1, 2, 6	idfu1st 17970	. 2 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐶)))
8		idfu2nda.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
9	1, 3, 8	idfu1stalem 50011	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
10	9	reseq2d 5976	. 2 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘𝐶)))
11	7, 10	eqtr4d 2800	1 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 I cid 5553 ↾ cres 5661 ‘cfv 6537 (class class class)co 7416 1^st c1st 7987 Basecbs 17303 Catccat 17754 Func cfunc 17945 id_funccidfu 17946

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7989 df-2nd 7990 df-map 8831 df-ixp 8908 df-cat 17758 df-cid 17759 df-homf 17760 df-func 17949 df-idfu 17950

This theorem is used by: imaidfu2lem 50020 imaidfu 50021 imaidfu2 50022 idsubc 50071

W3C validator