Theorem cofidfth 49073

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49038, 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 2730	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1	adantr 480	. . . . 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 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
9		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
11		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
13	2, 3, 4, 6, 8, 9, 10, 11, 12	cofidf2 49037	. . . 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 494	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
15	14	ralrimivva 3182	. 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 17885	. 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 583	1 ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  ⟨cop 4603   class class class wbr 5115  –1-1→wf1 6516  –onto→wfo 6517  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  Hom chom 17237   Func cfunc 17822  id_funccidfu 17823   ∘_func ccofu 17824   Faith cfth 17873

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 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-func 17826  df-idfu 17827  df-cofu 17828  df-fth 17875

This theorem is referenced by:  uobeqw  49125  uobeq  49126