idemb - Mathbox for Zhi Wang

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Theorem idemb 50606

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 50605	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
3	1	eleq1i 2855	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) ↔ (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
4		idfurcl 50545	. . . . 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 50546	. . . 4 ⊢ (𝐶 ∈ Cat → (1^st ‘𝐼):(Base‘𝐶)⟶1-1onto(Base‘𝐶))
8		dff1o4 6863	. . . . 5 ⊢ ((1^st ‘𝐼):(Base‘𝐶)⟶1-1onto(Base‘𝐶) ↔ ((1^st ‘𝐼) Fn (Base‘𝐶) ∧ ^◡(1^st ‘𝐼) Fn (Base‘𝐶)))
9	8	simprbi 503	. . . 4 ⊢ ((1^st ‘𝐼):(Base‘𝐶)⟶1-1onto(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 6670	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ^◡(1^st ‘𝐼))
12	2, 11	jca 521	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ^◡ccnv 5657 Fun wfun 6560 Fn wfn 6561 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 Basecbs 17431 Catccat 17888 Func cfunc 18079 id_funccidfu 18080 Faith cfth 18130

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-map 8887 df-ixp 8964 df-cat 17892 df-cid 17893 df-homf 17894 df-func 18083 df-idfu 18084 df-fth 18132

This theorem is used by: (None)

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