Theorem thincfth 49639

Description: A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
thincfth.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincfth.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)

Assertion

Ref	Expression
thincfth	⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

Proof of Theorem thincfth

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

Step	Hyp	Ref	Expression
1		thincfth.f	. 2 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
2		thincfth.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
4		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
5		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
6		eqid 2734	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	3, 4, 5, 6, 7	thincmo 49615	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
9		eqid 2734	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
11	6, 7, 9, 10, 4, 5	funcf2 17790	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
12		f1mo 49040	. . . 4 ⊢ ((∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
13	8, 11, 12	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
14	13	ralrimivva 3177	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
15	6, 7, 9	isfth2 17839	. 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))))
16	1, 14, 15	sylanbrc 583	1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∃*wmo 2535  ∀wral 3049   class class class wbr 5096  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Hom chom 17186   Func cfunc 17776   Faith cfth 17827  ThinCatcthinc 49604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-ixp 8834  df-func 17780  df-fth 17829  df-thinc 49605

This theorem is referenced by:  thincciso  49640