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| Mirrors > Home > MPE Home > Th. List > isepi | Structured version Visualization version GIF version | ||
| Description: Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| isepi.b | ⊢ 𝐵 = (Base‘𝐶) | 
| isepi.h | ⊢ 𝐻 = (Hom ‘𝐶) | 
| isepi.o | ⊢ · = (comp‘𝐶) | 
| isepi.e | ⊢ 𝐸 = (Epi‘𝐶) | 
| isepi.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| isepi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| isepi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| isepi | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
| 2 | isepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 3 | 1, 2 | oppcbas 17762 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) | 
| 4 | eqid 2736 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
| 5 | eqid 2736 | . . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶)) | |
| 6 | eqid 2736 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
| 7 | isepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 8 | 1 | oppccat 17766 | . . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat) | 
| 9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat) | 
| 10 | isepi.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 11 | isepi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 12 | 3, 4, 5, 6, 9, 10, 11 | ismon 17778 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋) ↔ (𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔))))) | 
| 13 | isepi.e | . . . 4 ⊢ 𝐸 = (Epi‘𝐶) | |
| 14 | 1, 7, 6, 13 | oppcmon 17783 | . . 3 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) | 
| 15 | 14 | eleq2d 2826 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐸𝑌))) | 
| 16 | isepi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 17 | 16, 1 | oppchom 17759 | . . . . 5 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) | 
| 18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌)) | 
| 19 | 18 | eleq2d 2826 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐻𝑌))) | 
| 20 | 16, 1 | oppchom 17759 | . . . . . . . 8 ⊢ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) = (𝑌𝐻𝑧) | 
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧(Hom ‘(oppCat‘𝐶))𝑌) = (𝑌𝐻𝑧)) | 
| 22 | isepi.o | . . . . . . . 8 ⊢ · = (comp‘𝐶) | |
| 23 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | |
| 24 | 10 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑌 ∈ 𝐵) | 
| 25 | 11 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵) | 
| 26 | 2, 22, 1, 23, 24, 25 | oppcco 17761 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔) = (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)) | 
| 27 | 21, 26 | mpteq12dv 5232 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔)) = (𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))) | 
| 28 | 27 | cnveqd 5885 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ◡(𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔)) = ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))) | 
| 29 | 28 | funeqd 6587 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (Fun ◡(𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)))) | 
| 30 | 29 | ralbidva 3175 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔)) ↔ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)))) | 
| 31 | 19, 30 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧(Hom ‘(oppCat‘𝐶))𝑌) ↦ (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔))) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) | 
| 32 | 12, 15, 31 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 〈cop 4631 ↦ cmpt 5224 ◡ccnv 5683 Fun wfun 6554 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Hom chom 17309 compcco 17310 Catccat 17708 oppCatcoppc 17755 Monocmon 17773 Epicepi 17774 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-hom 17322 df-cco 17323 df-cat 17712 df-cid 17713 df-oppc 17756 df-mon 17775 df-epi 17776 | 
| This theorem is referenced by: isepi2 17786 epihom 17787 | 
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