Theorem cofidfth 49515

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49474, 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 2737	. . . . 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 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
11		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐷))
12		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
13	2, 3, 4, 6, 8, 9, 10, 11, 12	cofidf2 49473	. . . 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 3181	. 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 17853	. 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 584	1 ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4588   class class class wbr 5100  –1-1→wf1 6497  –onto→wfo 6498  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200   Func cfunc 17790  id_funccidfu 17791   ∘_func ccofu 17792   Faith cfth 17841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-func 17794  df-idfu 17795  df-cofu 17796  df-fth 17843

This theorem is referenced by:  uobeqw  49572  uobeq  49573