Theorem cofidfth 50096

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 50055, this theorem proves that 𝐹 is an embedding (a faithful functor injective on objects, remark 3.28(1) of [Adamek] p. 34). (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofidfth.i	⊢ 𝐼 = (idfunc‘𝐷)
cofidfth.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
cofidfth.k	⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
cofidfth.o	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)

Assertion

Ref	Expression
cofidfth	⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)

Proof of Theorem cofidfth

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

Step	Hyp	Ref	Expression
1		cofidfth.f	. 2 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
2		cofidfth.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
3		eqid 2762	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹(𝐷 Func 𝐸)𝐺)
5		cofidfth.k	. . . . . 6 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐾(𝐸 Func 𝐷)𝐿)
7		cofidfth.o	. . . . . 6 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
9		eqid 2762	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2762	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
11		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
13	2, 3, 4, 6, 8, 9, 10, 11, 12	cofidf2 50054	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → ((𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)) ∧ ((𝐹‘𝑥)𝐿(𝐹‘𝑦)):((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦))–onto→(𝑥(Hom ‘𝐷)𝑦)))
14	13	simpld 500	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
15	14	ralrimivva 3207	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
16	3, 9, 10	isfth2 18012	. 2 ⊢ (𝐹(𝐷 Faith 𝐸)𝐺 ↔ (𝐹(𝐷 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ (Base‘𝐷)∀𝑦 ∈ (Base‘𝐷)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦))))
17	1, 15, 16	sylanbrc 595	1 ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ⟨cop 4593   class class class wbr 5107  –1-1→wf1 6534  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Hom chom 17359   Func cfunc 17949  id_funccidfu 17950   ∘_func ccofu 17951   Faith cfth 18000

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ixp 8909  df-func 17953  df-idfu 17954  df-cofu 17955  df-fth 18002

This theorem is used by:  uobeqw  50153  uobeq  50154