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Mirrors > Home > MPE Home > Th. List > oppcepi | Structured version Visualization version GIF version |
Description: An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcmon.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcmon.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppcepi.e | ⊢ 𝐸 = (Epi‘𝑂) |
oppcepi.m | ⊢ 𝑀 = (Mono‘𝐶) |
Ref | Expression |
---|---|
oppcepi | ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcepi.m | . . . 4 ⊢ 𝑀 = (Mono‘𝐶) | |
2 | oppcmon.o | . . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶) | |
3 | 2 | 2oppchomf 16773 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
5 | 2 | 2oppccomf 16774 | . . . . . 6 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → (compf‘𝐶) = (compf‘(oppCat‘𝑂))) |
7 | oppcmon.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | 2 | oppccat 16771 | . . . . . . 7 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | eqid 2778 | . . . . . . 7 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
11 | 10 | oppccat 16771 | . . . . . 6 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat) |
12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat) |
13 | 4, 6, 7, 12 | monpropd 16786 | . . . 4 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘(oppCat‘𝑂))) |
14 | 1, 13 | syl5eq 2826 | . . 3 ⊢ (𝜑 → 𝑀 = (Mono‘(oppCat‘𝑂))) |
15 | 14 | oveqd 6941 | . 2 ⊢ (𝜑 → (𝑌𝑀𝑋) = (𝑌(Mono‘(oppCat‘𝑂))𝑋)) |
16 | eqid 2778 | . . 3 ⊢ (Mono‘(oppCat‘𝑂)) = (Mono‘(oppCat‘𝑂)) | |
17 | oppcepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝑂) | |
18 | 10, 9, 16, 17 | oppcmon 16787 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝑂))𝑋) = (𝑋𝐸𝑌)) |
19 | 15, 18 | eqtr2d 2815 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 Catccat 16714 Homf chomf 16716 compfccomf 16717 oppCatcoppc 16760 Monocmon 16777 Epicepi 16778 |
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 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
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 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-hom 16366 df-cco 16367 df-cat 16718 df-cid 16719 df-homf 16720 df-comf 16721 df-oppc 16761 df-mon 16779 df-epi 16780 |
This theorem is referenced by: (None) |
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