Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6663 | . . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶)) | |
4 | oppcmon.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | 3, 4 | syl6eqr 2871 | . . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂) |
6 | 5 | fveq2d 6667 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂)) |
7 | oppcmon.m | . . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂) | |
8 | 6, 7 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀) |
9 | 8 | tposeqd 7884 | . . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀) |
10 | df-epi 16989 | . . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | |
11 | 7 | fvexi 6677 | . . . . . . 7 ⊢ 𝑀 ∈ V |
12 | 11 | tposex 7915 | . . . . . 6 ⊢ tpos 𝑀 ∈ V |
13 | 9, 10, 12 | fvmpt 6761 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀) |
15 | 1, 14 | syl5eq 2865 | . . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀) |
16 | 15 | oveqd 7162 | . 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋)) |
17 | ovtpos 7896 | . 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌) | |
18 | 16, 17 | syl6req 2870 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 tpos ctpos 7880 Catccat 16923 oppCatcoppc 16969 Monocmon 16986 Epicepi 16987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-ov 7148 df-tpos 7881 df-epi 16989 |
This theorem is referenced by: oppcepi 16997 isepi 16998 epii 17001 sectepi 17042 episect 17043 fthepi 17186 |
Copyright terms: Public domain | W3C validator |