imaidfu2lem - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > imaidfu2lem		Structured version Visualization version GIF version

Theorem imaidfu2lem 49607

Description: Lemma for imaidfu2 49609. (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 2740	. . . 4 ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝐷))
4	1, 2, 3	idfu1sta 49599	. . 3 ⊢ (𝜑 → (1^st ‘𝐼) = ( I ↾ (Base‘𝐷)))
5	4	imaeq1d 6012	. 2 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (( I ↾ (Base‘𝐷)) “ (Base‘𝐷)))
6		ssid 3937	. . 3 ⊢ (Base‘𝐷) ⊆ (Base‘𝐷)
7		resiima 6029	. . 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 2790	1 ⊢ (𝜑 → ((1^st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ⊆ wss 3883 I cid 5513 ↾ cres 5621 “ cima 5622 ‘cfv 6486 (class class class)co 7357 1^st c1st 7930 Basecbs 17171 Func cfunc 17813 id_funccidfu 17814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7932 df-2nd 7933 df-map 8766 df-ixp 8837 df-cat 17626 df-cid 17627 df-homf 17628 df-func 17817 df-idfu 17818

This theorem is referenced by: idsubc 49658 idfullsubc 49659

W3C validator