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Mirrors > Home > MPE Home > Th. List > sectepi | Structured version Visualization version GIF version |
Description: If 𝐹 is a section of 𝐺, then 𝐺 is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
sectepi.b | ⊢ 𝐵 = (Base‘𝐶) |
sectepi.e | ⊢ 𝐸 = (Epi‘𝐶) |
sectepi.s | ⊢ 𝑆 = (Sect‘𝐶) |
sectepi.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
sectepi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
sectepi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
sectepi.1 | ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) |
Ref | Expression |
---|---|
sectepi | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐸𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | sectepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 16982 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2821 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
5 | eqid 2821 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
6 | sectepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | 1 | oppccat 16986 | . . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat) |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat) |
9 | sectepi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | sectepi.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | sectepi.1 | . . . 4 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) | |
12 | sectepi.s | . . . . 5 ⊢ 𝑆 = (Sect‘𝐶) | |
13 | 2, 1, 6, 9, 10, 12, 5 | oppcsect 17042 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
14 | 11, 13 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
15 | 3, 4, 5, 8, 9, 10, 14 | sectmon 17046 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Mono‘(oppCat‘𝐶))𝑌)) |
16 | sectepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝐶) | |
17 | 1, 6, 4, 16 | oppcmon 17002 | . 2 ⊢ (𝜑 → (𝑋(Mono‘(oppCat‘𝐶))𝑌) = (𝑌𝐸𝑋)) |
18 | 15, 17 | eleqtrd 2915 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Catccat 16929 oppCatcoppc 16975 Monocmon 16992 Epicepi 16993 Sectcsect 17008 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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 df-sect 17011 |
This theorem is referenced by: (None) |
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