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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 6906 | . . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶)) | |
| 4 | oppcmon.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂) |
| 6 | 5 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂)) |
| 7 | oppcmon.m | . . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀) |
| 9 | 8 | tposeqd 8254 | . . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀) |
| 10 | df-epi 17775 | . . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | |
| 11 | 7 | fvexi 6920 | . . . . . . 7 ⊢ 𝑀 ∈ V |
| 12 | 11 | tposex 8285 | . . . . . 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 2789 | . . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀) |
| 16 | 15 | oveqd 7448 | . 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋)) |
| 17 | ovtpos 8266 | . 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌) | |
| 18 | 16, 17 | eqtr2di 2794 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 Catccat 17707 oppCatcoppc 17754 Monocmon 17772 Epicepi 17773 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-tpos 8251 df-epi 17775 |
| This theorem is referenced by: oppcepi 17783 isepi 17784 epii 17787 sectepi 17828 episect 17829 fthepi 17975 |
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