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Mirrors > Home > MPE Home > Th. List > oppcmon | Structured version Visualization version GIF version |
Description: A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcmon.o | β’ π = (oppCatβπΆ) |
oppcmon.c | β’ (π β πΆ β Cat) |
oppcmon.m | β’ π = (Monoβπ) |
oppcmon.e | β’ πΈ = (EpiβπΆ) |
Ref | Expression |
---|---|
oppcmon | β’ (π β (πππ) = (ππΈπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcmon.e | . . . 4 β’ πΈ = (EpiβπΆ) | |
2 | oppcmon.c | . . . . 5 β’ (π β πΆ β Cat) | |
3 | fveq2 6897 | . . . . . . . . . 10 β’ (π = πΆ β (oppCatβπ) = (oppCatβπΆ)) | |
4 | oppcmon.o | . . . . . . . . . 10 β’ π = (oppCatβπΆ) | |
5 | 3, 4 | eqtr4di 2786 | . . . . . . . . 9 β’ (π = πΆ β (oppCatβπ) = π) |
6 | 5 | fveq2d 6901 | . . . . . . . 8 β’ (π = πΆ β (Monoβ(oppCatβπ)) = (Monoβπ)) |
7 | oppcmon.m | . . . . . . . 8 β’ π = (Monoβπ) | |
8 | 6, 7 | eqtr4di 2786 | . . . . . . 7 β’ (π = πΆ β (Monoβ(oppCatβπ)) = π) |
9 | 8 | tposeqd 8234 | . . . . . 6 β’ (π = πΆ β tpos (Monoβ(oppCatβπ)) = tpos π) |
10 | df-epi 17713 | . . . . . 6 β’ Epi = (π β Cat β¦ tpos (Monoβ(oppCatβπ))) | |
11 | 7 | fvexi 6911 | . . . . . . 7 β’ π β V |
12 | 11 | tposex 8265 | . . . . . 6 β’ tpos π β V |
13 | 9, 10, 12 | fvmpt 7005 | . . . . 5 β’ (πΆ β Cat β (EpiβπΆ) = tpos π) |
14 | 2, 13 | syl 17 | . . . 4 β’ (π β (EpiβπΆ) = tpos π) |
15 | 1, 14 | eqtrid 2780 | . . 3 β’ (π β πΈ = tpos π) |
16 | 15 | oveqd 7437 | . 2 β’ (π β (ππΈπ) = (πtpos ππ)) |
17 | ovtpos 8246 | . 2 β’ (πtpos ππ) = (πππ) | |
18 | 16, 17 | eqtr2di 2785 | 1 β’ (π β (πππ) = (ππΈπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 tpos ctpos 8230 Catccat 17643 oppCatcoppc 17690 Monocmon 17710 Epicepi 17711 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-fv 6556 df-ov 7423 df-tpos 8231 df-epi 17713 |
This theorem is referenced by: oppcepi 17721 isepi 17722 epii 17725 sectepi 17766 episect 17767 fthepi 17916 |
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