Theorem idfth 49621

Description: The inclusion functor is a faithful functor. (Contributed by Zhi Wang, 10-Nov-2025.)

Hypothesis

Ref	Expression
idfth.i	⊢ 𝐼 = (idfunc‘𝐶)

Assertion

Ref	Expression
idfth	⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))

Proof of Theorem idfth

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17818	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		1st2nd 7981	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐼 ∈ (𝐷 Func 𝐸)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
3	1, 2	mpan 691	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
4		id 22	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 49539	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
6		f1oi 6807	. . . . . . . 8 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
7		dff1o3 6775	. . . . . . . 8 ⊢ (( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦) ↔ (( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–onto→(𝑥(Hom ‘𝐷)𝑦) ∧ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦))))
8	6, 7	mpbi 230	. . . . . . 7 ⊢ (( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–onto→(𝑥(Hom ‘𝐷)𝑦) ∧ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
9	8	simpri 485	. . . . . 6 ⊢ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦))
10		idfth.i	. . . . . . . . 9 ⊢ 𝐼 = (idfunc‘𝐶)
11		simpl 482	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐼 ∈ (𝐷 Func 𝐸))
12		eqidd 2736	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐷))
13		simprl 771	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
14		simprr 773	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
15		eqidd 2736	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
16	10, 11, 12, 13, 14, 15	idfu2nda 49566	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
17	16	cnveqd 5819	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ◡(𝑥(2nd ‘𝐼)𝑦) = ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
18	17	funeqd 6509	. . . . . 6 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Fun ◡(𝑥(2nd ‘𝐼)𝑦) ↔ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦))))
19	9, 18	mpbiri 258	. . . . 5 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → Fun ◡(𝑥(2nd ‘𝐼)𝑦))
20	19	ralrimivva 3178	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)Fun ◡(𝑥(2nd ‘𝐼)𝑦))
21		eqid 2735	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
22	21	isfth 17872	. . . 4 ⊢ ((1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼) ↔ ((1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼) ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)Fun ◡(𝑥(2nd ‘𝐼)𝑦)))
23	5, 20, 22	sylanbrc 584	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼))
24		df-br 5075	. . 3 ⊢ ((1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
25	23, 24	sylib 218	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
26	3, 25	eqeltrd 2835	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049  ⟨cop 4563   class class class wbr 5074   I cid 5514  ^◡ccnv 5619   ↾ cres 5622  Rel wrel 5625  Fun wfun 6481  –onto→wfo 6485  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17168  Hom chom 17220   Func cfunc 17810  id_funccidfu 17811   Faith cfth 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8764  df-ixp 8835  df-cat 17623  df-cid 17624  df-homf 17625  df-func 17814  df-idfu 17815  df-fth 17863

This theorem is referenced by:  idemb  49622