imaidfu2lem - Mathbox for Zhi Wang

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Theorem imaidfu2lem 49963

Description: Lemma for imaidfu2 49965. (Contributed by Zhi Wang, 10-Nov-2025.)

**Hypotheses**
Ref	Expression
imaidfu.i	⊢ 𝐼 = (id_func‘𝐶)
imaidfu.d	⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))

**Assertion**
Ref	Expression
imaidfu2lem	⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))

**Proof of Theorem imaidfu2lem**
Step	Hyp	Ref	Expression
1		imaidfu.i	. . . 4 ⊢ 𝐼 = (id_func‘𝐶)
2		imaidfu.d	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))
3		eqidd 2766	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49955	. . 3 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	imaeq1d 6063	. 2 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
6		ssid 3960	. . 3 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
7		resiima 6080	. . 3 ⊢ ((Base‘𝐷) ⊆ (Base‘𝐷) → (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)) = (Base‘𝐷))
8	6, 7	ax-mp 5	. 2 ⊢ (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)) = (Base‘𝐷)
9	5, 8	eqtrdi 2816	1 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 I cid 5557 ↾ cres 5665 “ cima 5666 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 Basecbs 17295 Func cfunc 17937 id_funccidfu 17938

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-map 8832 df-ixp 8902 df-cat 17750 df-cid 17751 df-homf 17752 df-func 17941 df-idfu 17942

This theorem is used by: idsubc 50014 idfullsubc 50015

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