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Mirrors > Home > MPE Home > Th. List > fthepi | Structured version Visualization version GIF version |
Description: A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fthmon.b | ⊢ 𝐵 = (Base‘𝐶) |
fthmon.h | ⊢ 𝐻 = (Hom ‘𝐶) |
fthmon.f | ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
fthmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
fthmon.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
fthmon.r | ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) |
fthepi.e | ⊢ 𝐸 = (Epi‘𝐶) |
fthepi.p | ⊢ 𝑃 = (Epi‘𝐷) |
fthepi.1 | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
Ref | Expression |
---|---|
fthepi | ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐸𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | fthmon.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16763 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2778 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2778 | . . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷) | |
6 | fthmon.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
7 | 1, 5, 6 | fthoppc 16968 | . . 3 ⊢ (𝜑 → 𝐹((oppCat‘𝐶) Faith (oppCat‘𝐷))tpos 𝐺) |
8 | fthmon.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | fthmon.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | fthmon.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) | |
11 | fthmon.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
12 | 11, 1 | oppchom 16760 | . . . 4 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
13 | 10, 12 | syl6eleqr 2870 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋)) |
14 | eqid 2778 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
15 | eqid 2778 | . . 3 ⊢ (Mono‘(oppCat‘𝐷)) = (Mono‘(oppCat‘𝐷)) | |
16 | ovtpos 7649 | . . . . . 6 ⊢ (𝑌tpos 𝐺𝑋) = (𝑋𝐺𝑌) | |
17 | 16 | fveq1i 6447 | . . . . 5 ⊢ ((𝑌tpos 𝐺𝑋)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅) |
18 | fthepi.1 | . . . . 5 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) | |
19 | 17, 18 | syl5eqel 2863 | . . . 4 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
20 | fthfunc 16952 | . . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
21 | 20 | ssbri 4931 | . . . . . . . . 9 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺) |
22 | 6, 21 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
23 | df-br 4887 | . . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
24 | 22, 23 | sylib 210 | . . . . . . 7 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
25 | funcrcl 16908 | . . . . . . 7 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
27 | 26 | simprd 491 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat) |
28 | fthepi.p | . . . . 5 ⊢ 𝑃 = (Epi‘𝐷) | |
29 | 5, 27, 15, 28 | oppcmon 16783 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋)) = ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
30 | 19, 29 | eleqtrrd 2862 | . . 3 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋))) |
31 | 3, 4, 7, 8, 9, 13, 14, 15, 30 | fthmon 16972 | . 2 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋)) |
32 | 26 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
33 | fthepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝐶) | |
34 | 1, 32, 14, 33 | oppcmon 16783 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
35 | 31, 34 | eleqtrd 2861 | 1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 〈cop 4404 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 tpos ctpos 7633 Basecbs 16255 Hom chom 16349 Catccat 16710 oppCatcoppc 16756 Monocmon 16773 Epicepi 16774 Func cfunc 16899 Faith cfth 16948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-hom 16362 df-cco 16363 df-cat 16714 df-cid 16715 df-oppc 16757 df-mon 16775 df-epi 16776 df-func 16903 df-fth 16950 |
This theorem is referenced by: (None) |
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