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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 2726 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | fthmon.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17724 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2726 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2726 | . . . 4 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷) | |
6 | fthmon.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
7 | 1, 5, 6 | fthoppc 17937 | . . 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 17721 | . . . 4 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
13 | 10, 12 | eleqtrrdi 2837 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋)) |
14 | eqid 2726 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
15 | eqid 2726 | . . 3 ⊢ (Mono‘(oppCat‘𝐷)) = (Mono‘(oppCat‘𝐷)) | |
16 | ovtpos 8245 | . . . . . 6 ⊢ (𝑌tpos 𝐺𝑋) = (𝑋𝐺𝑌) | |
17 | 16 | fveq1i 6891 | . . . . 5 ⊢ ((𝑌tpos 𝐺𝑋)‘𝑅) = ((𝑋𝐺𝑌)‘𝑅) |
18 | fthepi.1 | . . . . 5 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) | |
19 | 17, 18 | eqeltrid 2830 | . . . 4 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
20 | fthfunc 17921 | . . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
21 | 20 | ssbri 5188 | . . . . . . . . 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 5144 | . . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
24 | 22, 23 | sylib 217 | . . . . . . 7 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
25 | funcrcl 17874 | . . . . . . 7 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
27 | 26 | simprd 494 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat) |
28 | fthepi.p | . . . . 5 ⊢ 𝑃 = (Epi‘𝐷) | |
29 | 5, 27, 15, 28 | oppcmon 17746 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋)) = ((𝐹‘𝑋)𝑃(𝐹‘𝑌))) |
30 | 19, 29 | eleqtrrd 2829 | . . 3 ⊢ (𝜑 → ((𝑌tpos 𝐺𝑋)‘𝑅) ∈ ((𝐹‘𝑌)(Mono‘(oppCat‘𝐷))(𝐹‘𝑋))) |
31 | 3, 4, 7, 8, 9, 13, 14, 15, 30 | fthmon 17941 | . 2 ⊢ (𝜑 → 𝑅 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋)) |
32 | 26 | simpld 493 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
33 | fthepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝐶) | |
34 | 1, 32, 14, 33 | oppcmon 17746 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
35 | 31, 34 | eleqtrd 2828 | 1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐸𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 〈cop 4629 class class class wbr 5143 ‘cfv 6543 (class class class)co 7413 tpos ctpos 8229 Basecbs 17205 Hom chom 17269 Catccat 17669 oppCatcoppc 17716 Monocmon 17736 Epicepi 17737 Func cfunc 17865 Faith cfth 17917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-z 12602 df-dec 12721 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-hom 17282 df-cco 17283 df-cat 17673 df-cid 17674 df-oppc 17717 df-mon 17738 df-epi 17739 df-func 17869 df-fth 17919 |
This theorem is referenced by: (None) |
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