Theorem thincfth 50918

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 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
4		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
5		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
6		eqid 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	3, 4, 5, 6, 7	thincmo 50894	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
9		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
11	6, 7, 9, 10, 4, 5	funcf2 18093	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
12		f1mo 50295	. . . 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 596	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶1-1((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
14	13	ralrimivva 3209	. 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 18142	. 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 595	1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∃*wmo 2565  ∀wral 3080   class class class wbr 5108  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Hom chom 17483   Func cfunc 18079   Faith cfth 18130  ThinCatcthinc 50883

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-ixp 8964  df-func 18083  df-fth 18132  df-thinc 50884

This theorem is used by:  thincciso  50919