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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 6846 | . . . . . . . . . 10 β’ (π = πΆ β (oppCatβπ) = (oppCatβπΆ)) | |
4 | oppcmon.o | . . . . . . . . . 10 β’ π = (oppCatβπΆ) | |
5 | 3, 4 | eqtr4di 2791 | . . . . . . . . 9 β’ (π = πΆ β (oppCatβπ) = π) |
6 | 5 | fveq2d 6850 | . . . . . . . 8 β’ (π = πΆ β (Monoβ(oppCatβπ)) = (Monoβπ)) |
7 | oppcmon.m | . . . . . . . 8 β’ π = (Monoβπ) | |
8 | 6, 7 | eqtr4di 2791 | . . . . . . 7 β’ (π = πΆ β (Monoβ(oppCatβπ)) = π) |
9 | 8 | tposeqd 8164 | . . . . . 6 β’ (π = πΆ β tpos (Monoβ(oppCatβπ)) = tpos π) |
10 | df-epi 17622 | . . . . . 6 β’ Epi = (π β Cat β¦ tpos (Monoβ(oppCatβπ))) | |
11 | 7 | fvexi 6860 | . . . . . . 7 β’ π β V |
12 | 11 | tposex 8195 | . . . . . 6 β’ tpos π β V |
13 | 9, 10, 12 | fvmpt 6952 | . . . . 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 2785 | . . 3 β’ (π β πΈ = tpos π) |
16 | 15 | oveqd 7378 | . 2 β’ (π β (ππΈπ) = (πtpos ππ)) |
17 | ovtpos 8176 | . 2 β’ (πtpos ππ) = (πππ) | |
18 | 16, 17 | eqtr2di 2790 | 1 β’ (π β (πππ) = (ππΈπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 tpos ctpos 8160 Catccat 17552 oppCatcoppc 17599 Monocmon 17619 Epicepi 17620 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 df-ov 7364 df-tpos 8161 df-epi 17622 |
This theorem is referenced by: oppcepi 17630 isepi 17631 epii 17634 sectepi 17675 episect 17676 fthepi 17823 |
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