idemb - Mathbox for Zhi Wang

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Theorem idemb 49656

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 49655	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
3	1	eleq1i 2831	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) ↔ (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
4		idfurcl 49595	. . . . 5 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
5	3, 4	sylbi 218	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6		eqid 2740	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1, 6	idfu1stf1o 49596	. . . 4 ⊢ (𝐶 ∈ Cat → (1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶))
8		dff1o4 6782	. . . . 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 498	. . . 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 6593	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ^◡(1^st ‘𝐼))
12	2, 11	jca 516	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ^◡ccnv 5624 Fun wfun 6486 Fn wfn 6487 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 Basecbs 17177 Catccat 17628 Func cfunc 17819 id_funccidfu 17820 Faith cfth 17870

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 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

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 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-map 8772 df-ixp 8843 df-cat 17632 df-cid 17633 df-homf 17634 df-func 17823 df-idfu 17824 df-fth 17872

This theorem is referenced by: (None)

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