Theorem idfth 49151

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 17831	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		1st2nd 8021	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐼 ∈ (𝐷 Func 𝐸)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
3	1, 2	mpan 690	. 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 49069	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
6		f1oi 6841	. . . . . . . 8 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
7		dff1o3 6809	. . . . . . . 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 2731	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐷))
13		simprl 770	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
14		simprr 772	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
15		eqidd 2731	. . . . . . . . 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 49096	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
17	16	cnveqd 5842	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ◡(𝑥(2nd ‘𝐼)𝑦) = ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
18	17	funeqd 6541	. . . . . 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 3181	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)Fun ◡(𝑥(2nd ‘𝐼)𝑦))
21		eqid 2730	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
22	21	isfth 17885	. . . 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 583	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼))
24		df-br 5111	. . 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 2829	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟨cop 4598   class class class wbr 5110   I cid 5535  ^◡ccnv 5640   ↾ cres 5643  Rel wrel 5646  Fun wfun 6508  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17186  Hom chom 17238   Func cfunc 17823  id_funccidfu 17824   Faith cfth 17874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-ixp 8874  df-cat 17636  df-cid 17637  df-homf 17638  df-func 17827  df-idfu 17828  df-fth 17876

This theorem is referenced by:  idemb  49152