Theorem thincfth 50073

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 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
4		simprl 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
5		simprr 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
6		eqid 2762	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2762	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	3, 4, 5, 6, 7	thincmo 50049	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
9		eqid 2762	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
11	6, 7, 9, 10, 4, 5	funcf2 17901	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
12		f1mo 49474	. . . 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 593	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
14	13	ralrimivva 3205	. 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 17950	. 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 592	1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  ∃*wmo 2564  ∀wral 3076   class class class wbr 5100  ⟶wf 6517  –1-1→wf1 6518  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Hom chom 17297   Func cfunc 17887   Faith cfth 17938  ThinCatcthinc 50038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ixp 8880  df-func 17891  df-fth 17940  df-thinc 50039

This theorem is referenced by:  thincciso  50074