Theorem idfth 49656

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 17821	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		1st2nd 7982	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐼 ∈ (𝐷 Func 𝐸)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
3	1, 2	mpan 696	. 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 49574	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
6		f1oi 6806	. . . . . . . 8 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
7		dff1o3 6774	. . . . . . . 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 231	. . . . . . 7 ⊢ (( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–onto→(𝑥(Hom ‘𝐷)𝑦) ∧ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
9	8	simpri 486	. . . . . 6 ⊢ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦))
10		idfth.i	. . . . . . . . 9 ⊢ 𝐼 = (idfunc‘𝐶)
11		simpl 483	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐼 ∈ (𝐷 Func 𝐸))
12		eqidd 2740	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐷))
13		simprl 776	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
14		simprr 778	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
15		eqidd 2740	. . . . . . . . 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 49601	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
17	16	cnveqd 5818	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ◡(𝑥(2nd ‘𝐼)𝑦) = ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
18	17	funeqd 6508	. . . . . 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 259	. . . . 5 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → Fun ◡(𝑥(2nd ‘𝐼)𝑦))
20	19	ralrimivva 3182	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)Fun ◡(𝑥(2nd ‘𝐼)𝑦))
21		eqid 2739	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
22	21	isfth 17875	. . . 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 589	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼))
24		df-br 5074	. . 3 ⊢ ((1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
25	23, 24	sylib 219	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
26	3, 25	eqeltrd 2839	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ⟨cop 4562   class class class wbr 5073   I cid 5513  ^◡ccnv 5618   ↾ cres 5621  Rel wrel 5624  Fun wfun 6480  –onto→wfo 6484  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7357  1^st c1st 7930  2^nd c2nd 7931  Basecbs 17171  Hom chom 17223   Func cfunc 17813  id_funccidfu 17814   Faith cfth 17864

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 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

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 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-map 8766  df-ixp 8837  df-cat 17626  df-cid 17627  df-homf 17628  df-func 17817  df-idfu 17818  df-fth 17866

This theorem is referenced by:  idemb  49657