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Mirrors > Home > MPE Home > Th. List > subcssc | Structured version Visualization version GIF version |
Description: An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
subcixp.1 | β’ (π β π½ β (SubcatβπΆ)) |
subcssc.h | β’ π» = (Homf βπΆ) |
Ref | Expression |
---|---|
subcssc | β’ (π β π½ βcat π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcixp.1 | . . 3 β’ (π β π½ β (SubcatβπΆ)) | |
2 | subcssc.h | . . . 4 β’ π» = (Homf βπΆ) | |
3 | eqid 2730 | . . . 4 β’ (IdβπΆ) = (IdβπΆ) | |
4 | eqid 2730 | . . . 4 β’ (compβπΆ) = (compβπΆ) | |
5 | subcrcl 17767 | . . . . 5 β’ (π½ β (SubcatβπΆ) β πΆ β Cat) | |
6 | 1, 5 | syl 17 | . . . 4 β’ (π β πΆ β Cat) |
7 | eqidd 2731 | . . . 4 β’ (π β dom dom π½ = dom dom π½) | |
8 | 2, 3, 4, 6, 7 | issubc 17789 | . . 3 β’ (π β (π½ β (SubcatβπΆ) β (π½ βcat π» β§ βπ₯ β dom dom π½(((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β dom dom π½βπ§ β dom dom π½βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β©(compβπΆ)π§)π) β (π₯π½π§))))) |
9 | 1, 8 | mpbid 231 | . 2 β’ (π β (π½ βcat π» β§ βπ₯ β dom dom π½(((IdβπΆ)βπ₯) β (π₯π½π₯) β§ βπ¦ β dom dom π½βπ§ β dom dom π½βπ β (π₯π½π¦)βπ β (π¦π½π§)(π(β¨π₯, π¦β©(compβπΆ)π§)π) β (π₯π½π§)))) |
10 | 9 | simpld 493 | 1 β’ (π β π½ βcat π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βwral 3059 β¨cop 4633 class class class wbr 5147 dom cdm 5675 βcfv 6542 (class class class)co 7411 compcco 17213 Catccat 17612 Idccid 17613 Homf chomf 17614 βcat cssc 17758 Subcatcsubc 17760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-pm 8825 df-ixp 8894 df-ssc 17761 df-subc 17763 |
This theorem is referenced by: subcfn 17795 subcss1 17796 subcss2 17797 issubc3 17803 subsubc 17807 |
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