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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 17739 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))) |
5 | 2 | 2oppccomf 17740 | . . . . . 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 17737 | . . . . . . 7 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | eqid 2726 | . . . . . . 7 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂) | |
11 | 10 | oppccat 17737 | . . . . . 6 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat) |
12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat) |
13 | 4, 6, 7, 12 | monpropd 17753 | . . . 4 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘(oppCat‘𝑂))) |
14 | 1, 13 | eqtrid 2778 | . . 3 ⊢ (𝜑 → 𝑀 = (Mono‘(oppCat‘𝑂))) |
15 | 14 | oveqd 7441 | . 2 ⊢ (𝜑 → (𝑌𝑀𝑋) = (𝑌(Mono‘(oppCat‘𝑂))𝑋)) |
16 | eqid 2726 | . . 3 ⊢ (Mono‘(oppCat‘𝑂)) = (Mono‘(oppCat‘𝑂)) | |
17 | oppcepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝑂) | |
18 | 10, 9, 16, 17 | oppcmon 17754 | . 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝑂))𝑋) = (𝑋𝐸𝑌)) |
19 | 15, 18 | eqtr2d 2767 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Catccat 17677 Homf chomf 17679 compfccomf 17680 oppCatcoppc 17724 Monocmon 17744 Epicepi 17745 |
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 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-hom 17290 df-cco 17291 df-cat 17681 df-cid 17682 df-homf 17683 df-comf 17684 df-oppc 17725 df-mon 17746 df-epi 17747 |
This theorem is referenced by: (None) |
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