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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 2798 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | isepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16980 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2798 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2798 | . . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶)) | |
6 | eqid 2798 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
7 | isepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | 1 | oppccat 16984 | . . . 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 16995 | . 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 17000 | . . 3 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
15 | 14 | eleq2d 2875 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐸𝑌))) |
16 | isepi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
17 | 16, 1 | oppchom 16977 | . . . . 5 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌)) |
19 | 18 | eleq2d 2875 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐻𝑌))) |
20 | 16, 1 | oppchom 16977 | . . . . . . . 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 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | |
24 | 10 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑌 ∈ 𝐵) |
25 | 11 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵) |
26 | 2, 22, 1, 23, 24, 25 | oppcco 16979 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔) = (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)) |
27 | 21, 26 | mpteq12dv 5115 | . . . . . 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 5710 | . . . . 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 6346 | . . . 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 3161 | . . 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 633 | . 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 312 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 ↦ cmpt 5110 ◡ccnv 5518 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 oppCatcoppc 16973 Monocmon 16990 Epicepi 16991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-hom 16581 df-cco 16582 df-cat 16931 df-cid 16932 df-oppc 16974 df-mon 16992 df-epi 16993 |
This theorem is referenced by: isepi2 17003 epihom 17004 |
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