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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 2740 | . . . 4 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
2 | sectepi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17439 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
4 | eqid 2740 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
5 | eqid 2740 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
6 | sectepi.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | 1 | oppccat 17444 | . . . 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 17501 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
14 | 11, 13 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
15 | 3, 4, 5, 8, 9, 10, 14 | sectmon 17505 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Mono‘(oppCat‘𝐶))𝑌)) |
16 | sectepi.e | . . 3 ⊢ 𝐸 = (Epi‘𝐶) | |
17 | 1, 6, 4, 16 | oppcmon 17461 | . 2 ⊢ (𝜑 → (𝑋(Mono‘(oppCat‘𝐶))𝑌) = (𝑌𝐸𝑋)) |
18 | 15, 17 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐸𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 Basecbs 16923 Catccat 17384 oppCatcoppc 17431 Monocmon 17451 Epicepi 17452 Sectcsect 17467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-tpos 8034 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-hom 16997 df-cco 16998 df-cat 17388 df-cid 17389 df-oppc 17432 df-mon 17453 df-epi 17454 df-sect 17470 |
This theorem is referenced by: (None) |
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