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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 2821 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | isepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16982 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2821 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2821 | . . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶)) | |
6 | eqid 2821 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
7 | isepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | 1 | oppccat 16986 | . . . 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 16997 | . 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 17002 | . . 3 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
15 | 14 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐸𝑌))) |
16 | isepi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
17 | 16, 1 | oppchom 16979 | . . . . 5 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌)) |
19 | 18 | eleq2d 2898 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐻𝑌))) |
20 | 16, 1 | oppchom 16979 | . . . . . . . 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 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | |
24 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑌 ∈ 𝐵) |
25 | 11 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
26 | 2, 22, 1, 23, 24, 25 | oppcco 16981 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔) = (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)) |
27 | 21, 26 | mpteq12dv 5143 | . . . . . 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 5740 | . . . . 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 6371 | . . . 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 3196 | . . 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 311 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 〈cop 4566 ↦ cmpt 5138 ◡ccnv 5548 Fun wfun 6343 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 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-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-hom 16583 df-cco 16584 df-cat 16933 df-cid 16934 df-oppc 16976 df-mon 16994 df-epi 16995 |
This theorem is referenced by: isepi2 17005 epihom 17006 |
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