imaidfu2lem - Mathbox for Zhi Wang

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Theorem imaidfu2lem 49915

Description: Lemma for imaidfu2 49917. (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 2764	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49907	. . 3 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	imaeq1d 6061	. 2 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
6		ssid 3959	. . 3 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
7		resiima 6078	. . 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 2814	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 2143 ⊆ wss 3905 I cid 5555 ↾ cres 5663 “ cima 5664 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 Basecbs 17273 Func cfunc 17915 id_funccidfu 17916

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-map 8822 df-ixp 8892 df-cat 17728 df-cid 17729 df-homf 17730 df-func 17919 df-idfu 17920

This theorem is used by: idsubc 49966 idfullsubc 49967

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