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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 | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcmon.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppcmon.m | ⊢ 𝑀 = (Mono‘𝑂) |
oppcmon.e | ⊢ 𝐸 = (Epi‘𝐶) |
Ref | Expression |
---|---|
oppcmon | ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcmon.e | . . . 4 ⊢ 𝐸 = (Epi‘𝐶) | |
2 | oppcmon.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | fveq2 6645 | . . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶)) | |
4 | oppcmon.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | 3, 4 | eqtr4di 2851 | . . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂) |
6 | 5 | fveq2d 6649 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂)) |
7 | oppcmon.m | . . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂) | |
8 | 6, 7 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀) |
9 | 8 | tposeqd 7878 | . . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀) |
10 | df-epi 16993 | . . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | |
11 | 7 | fvexi 6659 | . . . . . . 7 ⊢ 𝑀 ∈ V |
12 | 11 | tposex 7909 | . . . . . 6 ⊢ tpos 𝑀 ∈ V |
13 | 9, 10, 12 | fvmpt 6745 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀) |
15 | 1, 14 | syl5eq 2845 | . . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀) |
16 | 15 | oveqd 7152 | . 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋)) |
17 | ovtpos 7890 | . 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌) | |
18 | 16, 17 | eqtr2di 2850 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 tpos ctpos 7874 Catccat 16927 oppCatcoppc 16973 Monocmon 16990 Epicepi 16991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ov 7138 df-tpos 7875 df-epi 16993 |
This theorem is referenced by: oppcepi 17001 isepi 17002 epii 17005 sectepi 17046 episect 17047 fthepi 17190 |
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