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Mirrors > Home > MPE Home > Th. List > epihom | Structured version Visualization version GIF version |
Description: An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
isepi.b | โข ๐ต = (Baseโ๐ถ) |
isepi.h | โข ๐ป = (Hom โ๐ถ) |
isepi.o | โข ยท = (compโ๐ถ) |
isepi.e | โข ๐ธ = (Epiโ๐ถ) |
isepi.c | โข (๐ โ ๐ถ โ Cat) |
isepi.x | โข (๐ โ ๐ โ ๐ต) |
isepi.y | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
epihom | โข (๐ โ (๐๐ธ๐) โ (๐๐ป๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isepi.b | . . . 4 โข ๐ต = (Baseโ๐ถ) | |
2 | isepi.h | . . . 4 โข ๐ป = (Hom โ๐ถ) | |
3 | isepi.o | . . . 4 โข ยท = (compโ๐ถ) | |
4 | isepi.e | . . . 4 โข ๐ธ = (Epiโ๐ถ) | |
5 | isepi.c | . . . 4 โข (๐ โ ๐ถ โ Cat) | |
6 | isepi.x | . . . 4 โข (๐ โ ๐ โ ๐ต) | |
7 | isepi.y | . . . 4 โข (๐ โ ๐ โ ๐ต) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isepi 17723 | . . 3 โข (๐ โ (๐ โ (๐๐ธ๐) โ (๐ โ (๐๐ป๐) โง โ๐ง โ ๐ต Fun โก(๐ โ (๐๐ป๐ง) โฆ (๐(โจ๐, ๐โฉ ยท ๐ง)๐))))) |
9 | simpl 482 | . . 3 โข ((๐ โ (๐๐ป๐) โง โ๐ง โ ๐ต Fun โก(๐ โ (๐๐ป๐ง) โฆ (๐(โจ๐, ๐โฉ ยท ๐ง)๐))) โ ๐ โ (๐๐ป๐)) | |
10 | 8, 9 | biimtrdi 252 | . 2 โข (๐ โ (๐ โ (๐๐ธ๐) โ ๐ โ (๐๐ป๐))) |
11 | 10 | ssrdv 3986 | 1 โข (๐ โ (๐๐ธ๐) โ (๐๐ป๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1534 โ wcel 2099 โwral 3058 โ wss 3947 โจcop 4635 โฆ cmpt 5231 โกccnv 5677 Fun wfun 6542 โcfv 6548 (class class class)co 7420 Basecbs 17180 Hom chom 17244 compcco 17245 Catccat 17644 Epicepi 17712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-hom 17257 df-cco 17258 df-cat 17648 df-cid 17649 df-oppc 17692 df-mon 17713 df-epi 17714 |
This theorem is referenced by: setcepi 18077 |
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