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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 6907 | . . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶)) | |
4 | oppcmon.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
5 | 3, 4 | eqtr4di 2793 | . . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂) |
6 | 5 | fveq2d 6911 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂)) |
7 | oppcmon.m | . . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂) | |
8 | 6, 7 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀) |
9 | 8 | tposeqd 8253 | . . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀) |
10 | df-epi 17779 | . . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | |
11 | 7 | fvexi 6921 | . . . . . . 7 ⊢ 𝑀 ∈ V |
12 | 11 | tposex 8284 | . . . . . 6 ⊢ tpos 𝑀 ∈ V |
13 | 9, 10, 12 | fvmpt 7016 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀) |
14 | 2, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀) |
15 | 1, 14 | eqtrid 2787 | . . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀) |
16 | 15 | oveqd 7448 | . 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋)) |
17 | ovtpos 8265 | . 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌) | |
18 | 16, 17 | eqtr2di 2792 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 tpos ctpos 8249 Catccat 17709 oppCatcoppc 17756 Monocmon 17776 Epicepi 17777 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ov 7434 df-tpos 8250 df-epi 17779 |
This theorem is referenced by: oppcepi 17787 isepi 17788 epii 17791 sectepi 17832 episect 17833 fthepi 17982 |
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