Theorem cofidfth 49791

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49750, 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 2765	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1	adantr 485	. . . . 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 485	. . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
9		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
11		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
13	2, 3, 4, 6, 8, 9, 10, 11, 12	cofidf2 49749	. . . 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 499	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐷)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))
15	14	ralrimivva 3208	. 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 17964	. 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 594	1 ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ⟨cop 4591   class class class wbr 5105  –1-1→wf1 6522  –onto→wfo 6523  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  Hom chom 17311   Func cfunc 17901  id_funccidfu 17902   ∘_func ccofu 17903   Faith cfth 17952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-func 17905  df-idfu 17906  df-cofu 17907  df-fth 17954

This theorem is referenced by:  uobeqw  49848  uobeq  49849