imaidfu2lem - Mathbox for Zhi Wang

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Theorem imaidfu2lem 49600

Description: Lemma for imaidfu2 49602. (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 2738	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49592	. . 3 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	imaeq1d 6020	. 2 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
6		ssid 3945	. . 3 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
7		resiima 6037	. . 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 2788	1 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 I cid 5520 ↾ cres 5628 “ cima 5629 ‘cfv 6494 (class class class)co 7362 1^st c1st 7935 Basecbs 17174 Func cfunc 17816 id_funccidfu 17817

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7937 df-2nd 7938 df-map 8770 df-ixp 8841 df-cat 17629 df-cid 17630 df-homf 17631 df-func 17820 df-idfu 17821

This theorem is referenced by: idsubc 49651 idfullsubc 49652

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