Theorem idfth 50188

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 17998	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		1st2nd 8033	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐼 ∈ (𝐷 Func 𝐸)) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
3	1, 2	mpan 703	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
4		id 23	. . . . 5 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 50106	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Func 𝐸)(2nd ‘𝐼))
6		f1oi 6851	. . . . . . . 8 ⊢ ( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–1-1-onto→(𝑥(Hom ‘𝐷)𝑦)
7		dff1o3 6819	. . . . . . . 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 233	. . . . . . 7 ⊢ (( I ↾ (𝑥(Hom ‘𝐷)𝑦)):(𝑥(Hom ‘𝐷)𝑦)–onto→(𝑥(Hom ‘𝐷)𝑦) ∧ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
9	8	simpri 491	. . . . . 6 ⊢ Fun ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦))
10		idfth.i	. . . . . . . . 9 ⊢ 𝐼 = (idfunc‘𝐶)
11		simpl 488	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐼 ∈ (𝐷 Func 𝐸))
12		eqidd 2761	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (Base‘𝐷) = (Base‘𝐷))
13		simprl 783	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
14		simprr 785	. . . . . . . . 9 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
15		eqidd 2761	. . . . . . . . 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 50133	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(2nd ‘𝐼)𝑦) = ( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
17	16	cnveqd 5849	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ◡(𝑥(2nd ‘𝐼)𝑦) = ◡( I ↾ (𝑥(Hom ‘𝐷)𝑦)))
18	17	funeqd 6549	. . . . . 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 261	. . . . 5 ⊢ ((𝐼 ∈ (𝐷 Func 𝐸) ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → Fun ◡(𝑥(2nd ‘𝐼)𝑦))
20	19	ralrimivva 3205	. . . 4 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)Fun ◡(𝑥(2nd ‘𝐼)𝑦))
21		eqid 2760	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
22	21	isfth 18052	. . . 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 595	. . 3 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼))
24		df-br 5103	. . 3 ⊢ ((1st ‘𝐼)(𝐷 Faith 𝐸)(2nd ‘𝐼) ↔ ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
25	23, 24	sylib 221	. 2 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ∈ (𝐷 Faith 𝐸))
26	3, 25	eqeltrd 2860	1 ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ⟨cop 4589   class class class wbr 5102   I cid 5541  ^◡ccnv 5646   ↾ cres 5649  Rel wrel 5652  Fun wfun 6521  –onto→wfo 6525  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17348  Hom chom 17400   Func cfunc 17990  id_funccidfu 17991   Faith cfth 18041

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-map 8827  df-ixp 8904  df-cat 17803  df-cid 17804  df-homf 17805  df-func 17994  df-idfu 17995  df-fth 18043

This theorem is used by:  idemb  50189