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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincfth | Structured version Visualization version GIF version |
Description: A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.) |
Ref | Expression |
---|---|
thincfth.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincfth.f | ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
Ref | Expression |
---|---|
thincfth | ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
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 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
5 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
6 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | eqid 2736 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
8 | 3, 4, 5, 6, 7 | thincmo 45926 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
9 | eqid 2736 | . . . . 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 17328 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))) |
12 | f1mo 45796 | . . . 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 587 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))) |
14 | 13 | ralrimivva 3102 | . 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 17376 | . 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 586 | 1 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ∃*wmo 2537 ∀wral 3051 class class class wbr 5039 ⟶wf 6354 –1-1→wf1 6355 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Func cfunc 17314 Faith cfth 17364 ThinCatcthinc 45916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-map 8488 df-ixp 8557 df-func 17318 df-fth 17366 df-thinc 45917 |
This theorem is referenced by: thincciso 45946 |
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