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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 6664 | . . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶)) | |
4 | oppcmon.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | 3, 4 | syl6eqr 2874 | . . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂) |
6 | 5 | fveq2d 6668 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂)) |
7 | oppcmon.m | . . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂) | |
8 | 6, 7 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀) |
9 | 8 | tposeqd 7889 | . . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀) |
10 | df-epi 16995 | . . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | |
11 | 7 | fvexi 6678 | . . . . . . 7 ⊢ 𝑀 ∈ V |
12 | 11 | tposex 7920 | . . . . . 6 ⊢ tpos 𝑀 ∈ V |
13 | 9, 10, 12 | fvmpt 6762 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀) |
15 | 1, 14 | syl5eq 2868 | . . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀) |
16 | 15 | oveqd 7167 | . 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋)) |
17 | ovtpos 7901 | . 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌) | |
18 | 16, 17 | syl6req 2873 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 tpos ctpos 7885 Catccat 16929 oppCatcoppc 16975 Monocmon 16992 Epicepi 16993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-ov 7153 df-tpos 7886 df-epi 16995 |
This theorem is referenced by: oppcepi 17003 isepi 17004 epii 17007 sectepi 17048 episect 17049 fthepi 17192 |
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