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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 16988 | . . 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 16992 | . . . 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 17003 | . 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 17008 | . . 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 16985 | . . . . 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 16985 | . . . . . . . 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 16987 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔) = (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)) |
27 | 21, 26 | mpteq12dv 5151 | . . . . . 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 5746 | . . . . 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 6377 | . . . 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 1537 ∈ wcel 2114 ∀wral 3138 〈cop 4573 ↦ cmpt 5146 ◡ccnv 5554 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 compcco 16577 Catccat 16935 oppCatcoppc 16981 Monocmon 16998 Epicepi 16999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-hom 16589 df-cco 16590 df-cat 16939 df-cid 16940 df-oppc 16982 df-mon 17000 df-epi 17001 |
This theorem is referenced by: isepi2 17011 epihom 17012 |
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