idemb - Mathbox for Zhi Wang

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Theorem idemb 49914

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 49913	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
3	1	eleq1i 2854	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) ↔ (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
4		idfurcl 49853	. . . . 5 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
5	3, 4	sylbi 220	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6		eqid 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1, 6	idfu1stf1o 49854	. . . 4 ⊢ (𝐶 ∈ Cat → (1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶))
8		dff1o4 6831	. . . . 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 502	. . . 4 ⊢ ((1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶) → ^◡(1^st ‘𝐼) Fn (Base‘𝐶))
10	5, 7, 9	3syl 19	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ^◡(1^st ‘𝐼) Fn (Base‘𝐶))
11	10	fnfund 6638	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ^◡(1^st ‘𝐼))
12	2, 11	jca 520	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ^◡ccnv 5662 Fun wfun 6532 Fn wfn 6533 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 Basecbs 17270 Catccat 17721 Func cfunc 17912 id_funccidfu 17913 Faith cfth 17963

This theorem was proved from 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 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem 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 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 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 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-map 8827 df-ixp 8897 df-cat 17725 df-cid 17726 df-homf 17727 df-func 17916 df-idfu 17917 df-fth 17965

This theorem is referenced by: (None)

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