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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 17433 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
5 | 2 | 2oppccomf 17434 | . . . . . 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 17431 | . . . . . . 7 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | eqid 2738 | . . . . . . 7 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
11 | 10 | oppccat 17431 | . . . . . 6 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat) |
12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat) |
13 | 4, 6, 7, 12 | monpropd 17447 | . . . 4 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘(oppCat‘𝑂))) |
14 | 1, 13 | eqtrid 2790 | . . 3 ⊢ (𝜑 → 𝑀 = (Mono‘(oppCat‘𝑂))) |
15 | 14 | oveqd 7294 | . 2 ⊢ (𝜑 → (𝑌𝑀𝑋) = (𝑌(Mono‘(oppCat‘𝑂))𝑋)) |
16 | eqid 2738 | . . 3 ⊢ (Mono‘(oppCat‘𝑂)) = (Mono‘(oppCat‘𝑂)) | |
17 | oppcepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝑂) | |
18 | 10, 9, 16, 17 | oppcmon 17448 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝑂))𝑋) = (𝑋𝐸𝑌)) |
19 | 15, 18 | eqtr2d 2779 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6435 (class class class)co 7277 Catccat 17371 Homf chomf 17373 compfccomf 17374 oppCatcoppc 17418 Monocmon 17438 Epicepi 17439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-hom 16984 df-cco 16985 df-cat 17375 df-cid 17376 df-homf 17377 df-comf 17378 df-oppc 17419 df-mon 17440 df-epi 17441 |
This theorem is referenced by: (None) |
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