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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 2760 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | fthmon.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16599 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2760 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2760 | . . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷) | |
6 | fthmon.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
7 | 1, 5, 6 | fthoppc 16804 | . . 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 16596 | . . . 4 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
13 | 10, 12 | syl6eleqr 2850 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋)) |
14 | eqid 2760 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
15 | eqid 2760 | . . 3 ⊢ (Mono‘(oppCat‘𝐷)) = (Mono‘(oppCat‘𝐷)) | |
16 | ovtpos 7537 | . . . . . 6 ⊢ (𝑌tpos 𝐺𝑋) = (𝑋𝐺𝑌) | |
17 | 16 | fveq1i 6354 | . . . . 5 ⊢ ((𝑌tpos 𝐺𝑋)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅) |
18 | fthepi.1 | . . . . 5 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) | |
19 | 17, 18 | syl5eqel 2843 | . . . 4 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
20 | fthfunc 16788 | . . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
21 | 20 | ssbri 4849 | . . . . . . . . 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 4805 | . . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
24 | 22, 23 | sylib 208 | . . . . . . 7 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
25 | funcrcl 16744 | . . . . . . 7 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
27 | 26 | simprd 482 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat) |
28 | fthepi.p | . . . . 5 ⊢ 𝑃 = (Epi‘𝐷) | |
29 | 5, 27, 15, 28 | oppcmon 16619 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋)) = ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
30 | 19, 29 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋))) |
31 | 3, 4, 7, 8, 9, 13, 14, 15, 30 | fthmon 16808 | . 2 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋)) |
32 | 26 | simpld 477 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
33 | fthepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝐶) | |
34 | 1, 32, 14, 33 | oppcmon 16619 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
35 | 31, 34 | eleqtrd 2841 | 1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 tpos ctpos 7521 Basecbs 16079 Hom chom 16174 Catccat 16546 oppCatcoppc 16592 Monocmon 16609 Epicepi 16610 Func cfunc 16735 Faith cfth 16784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-map 8027 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-hom 16188 df-cco 16189 df-cat 16550 df-cid 16551 df-oppc 16593 df-mon 16611 df-epi 16612 df-func 16739 df-fth 16786 |
This theorem is referenced by: (None) |
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