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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 6882 | . . . . . . . . . 10 β’ (π = πΆ β (oppCatβπ) = (oppCatβπΆ)) | |
4 | oppcmon.o | . . . . . . . . . 10 β’ π = (oppCatβπΆ) | |
5 | 3, 4 | eqtr4di 2782 | . . . . . . . . 9 β’ (π = πΆ β (oppCatβπ) = π) |
6 | 5 | fveq2d 6886 | . . . . . . . 8 β’ (π = πΆ β (Monoβ(oppCatβπ)) = (Monoβπ)) |
7 | oppcmon.m | . . . . . . . 8 β’ π = (Monoβπ) | |
8 | 6, 7 | eqtr4di 2782 | . . . . . . 7 β’ (π = πΆ β (Monoβ(oppCatβπ)) = π) |
9 | 8 | tposeqd 8210 | . . . . . 6 β’ (π = πΆ β tpos (Monoβ(oppCatβπ)) = tpos π) |
10 | df-epi 17683 | . . . . . 6 β’ Epi = (π β Cat β¦ tpos (Monoβ(oppCatβπ))) | |
11 | 7 | fvexi 6896 | . . . . . . 7 β’ π β V |
12 | 11 | tposex 8241 | . . . . . 6 β’ tpos π β V |
13 | 9, 10, 12 | fvmpt 6989 | . . . . 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 2776 | . . 3 β’ (π β πΈ = tpos π) |
16 | 15 | oveqd 7419 | . 2 β’ (π β (ππΈπ) = (πtpos ππ)) |
17 | ovtpos 8222 | . 2 β’ (πtpos ππ) = (πππ) | |
18 | 16, 17 | eqtr2di 2781 | 1 β’ (π β (πππ) = (ππΈπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 tpos ctpos 8206 Catccat 17613 oppCatcoppc 17660 Monocmon 17680 Epicepi 17681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-fv 6542 df-ov 7405 df-tpos 8207 df-epi 17683 |
This theorem is referenced by: oppcepi 17691 isepi 17692 epii 17695 sectepi 17736 episect 17737 fthepi 17886 |
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