Theorem thincfth 49942

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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat)
4		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
5		simprr 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
6		eqid 2739	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2739	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	3, 4, 5, 6, 7	thincmo 49918	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
9		eqid 2739	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
11	6, 7, 9, 10, 4, 5	funcf2 17826	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
12		f1mo 49343	. . . 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 590	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
14	13	ralrimivva 3182	. 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 17875	. 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 589	1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3053   class class class wbr 5072  ⟶wf 6481  –1-1→wf1 6482  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Hom chom 17222   Func cfunc 17812   Faith cfth 17863  ThinCatcthinc 49907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ixp 8836  df-func 17816  df-fth 17865  df-thinc 49908

This theorem is referenced by:  thincciso  49943