idemb - Mathbox for Zhi Wang

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Theorem idemb 49070

Description: The inclusion functor is an embedding. Remark 4.4(1) in [Adamek] p. 49. (Contributed by Zhi Wang, 16-Nov-2025.)

**Hypothesis**
Ref	Expression
idfth.i	⊢ 𝐼 = (id_func‘𝐶)

**Assertion**
Ref	Expression
idemb	⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

**Proof of Theorem idemb**
Step	Hyp	Ref	Expression
1		idfth.i	. . 3 ⊢ 𝐼 = (id_func‘𝐶)
2	1	idfth 49069	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
3	1	eleq1i 2820	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) ↔ (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
4		idfurcl 49015	. . . . 5 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
5	3, 4	sylbi 217	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6		eqid 2730	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1, 6	idfu1stf1o 49016	. . . 4 ⊢ (𝐶 ∈ Cat → (1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶))
8		dff1o4 6815	. . . . 5 ⊢ ((1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶) ↔ ((1^st ‘𝐼) Fn (Base‘𝐶) ∧ ^◡(1^st ‘𝐼) Fn (Base‘𝐶)))
9	8	simprbi 496	. . . 4 ⊢ ((1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶) → ^◡(1^st ‘𝐼) Fn (Base‘𝐶))
10	5, 7, 9	3syl 18	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ^◡(1^st ‘𝐼) Fn (Base‘𝐶))
11	10	fnfund 6627	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ^◡(1^st ‘𝐼))
12	2, 11	jca 511	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ^◡ccnv 5645 Fun wfun 6513 Fn wfn 6514 –1-1-onto→wf1o 6518 ‘cfv 6519 (class class class)co 7394 1^st c1st 7975 Basecbs 17185 Catccat 17631 Func cfunc 17822 id_funccidfu 17823 Faith cfth 17873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7977 df-2nd 7978 df-map 8805 df-ixp 8875 df-cat 17635 df-cid 17636 df-homf 17637 df-func 17826 df-idfu 17827 df-fth 17875

This theorem is referenced by: (None)

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