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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 2610 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | isepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16201 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2610 | . . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶)) | |
5 | eqid 2610 | . . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶)) | |
6 | eqid 2610 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
7 | isepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
8 | 1 | oppccat 16205 | . . . 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 16216 | . 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 16221 | . . 3 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
15 | 14 | eleq2d 2673 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐸𝑌))) |
16 | isepi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
17 | 16, 1 | oppchom 16198 | . . . . 5 ⊢ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑌(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑌)) |
19 | 18 | eleq2d 2673 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘(oppCat‘𝐶))𝑋) ↔ 𝐹 ∈ (𝑋𝐻𝑌))) |
20 | 16, 1 | oppchom 16198 | . . . . . . . 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 476 | . . . . . . . 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 16200 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹(〈𝑧, 𝑌〉(comp‘(oppCat‘𝐶))𝑋)𝑔) = (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹)) |
27 | 21, 26 | mpteq12dv 4663 | . . . . . 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 5220 | . . . . 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 5825 | . . . 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 2968 | . . 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 743 | . 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 297 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 ↦ cmpt 4643 ◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 oppCatcoppc 16194 Monocmon 16211 Epicepi 16212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-hom 15793 df-cco 15794 df-cat 16152 df-cid 16153 df-oppc 16195 df-mon 16213 df-epi 16214 |
This theorem is referenced by: isepi2 16224 epihom 16225 |
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