Theorem thincfth 48054

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 2728	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2728	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	3, 4, 5, 6, 7	thincmo 48035	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
9		eqid 2728	. . . . 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 17854	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
12		f1mo 47905	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
14	13	ralrimivva 3197	. 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 17904	. 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 582	1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∃*wmo 2528  ∀wral 3058   class class class wbr 5148  ⟶wf 6544  –1-1→wf1 6545  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  Hom chom 17244   Func cfunc 17840   Faith cfth 17892  ThinCatcthinc 48025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847  df-ixp 8917  df-func 17844  df-fth 17894  df-thinc 48026

This theorem is referenced by:  thincciso  48055