idemb - Mathbox for Zhi Wang

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Theorem idemb 49744

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 49743	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
3	1	eleq1i 2852	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) ↔ (id_func‘𝐶) ∈ (𝐷 Func 𝐸))
4		idfurcl 49683	. . . . 5 ⊢ ((id_func‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
5	3, 4	sylbi 219	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
6		eqid 2761	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1, 6	idfu1stf1o 49684	. . . 4 ⊢ (𝐶 ∈ Cat → (1^st ‘𝐼):(Base‘𝐶)–1-1-onto→(Base‘𝐶))
8		dff1o4 6811	. . . . 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 501	. . . 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 6618	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → Fun ^◡(1^st ‘𝐼))
12	2, 11	jca 519	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ^◡(1^st ‘𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ^◡ccnv 5644 Fun wfun 6511 Fn wfn 6512 –1-1-onto→wf1o 6516 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 Basecbs 17228 Catccat 17679 Func cfunc 17870 id_funccidfu 17871 Faith cfth 17921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-map 8805 df-ixp 8876 df-cat 17683 df-cid 17684 df-homf 17685 df-func 17874 df-idfu 17875 df-fth 17923

This theorem is referenced by: (None)

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