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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 2733 | . . . 4 β’ (IdβπΆ) = (IdβπΆ) | |
4 | eqid 2733 | . . . 4 β’ (compβπΆ) = (compβπΆ) | |
5 | subcrcl 17763 | . . . . 5 β’ (π½ β (SubcatβπΆ) β πΆ β Cat) | |
6 | 1, 5 | syl 17 | . . . 4 β’ (π β πΆ β Cat) |
7 | eqidd 2734 | . . . 4 β’ (π β dom dom π½ = dom dom π½) | |
8 | 2, 3, 4, 6, 7 | issubc 17785 | . . 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 496 | 1 β’ (π β π½ βcat π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 β¨cop 4635 class class class wbr 5149 dom cdm 5677 βcfv 6544 (class class class)co 7409 compcco 17209 Catccat 17608 Idccid 17609 Homf chomf 17610 βcat cssc 17754 Subcatcsubc 17756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-pm 8823 df-ixp 8892 df-ssc 17757 df-subc 17759 |
This theorem is referenced by: subcfn 17791 subcss1 17792 subcss2 17793 issubc3 17799 subsubc 17803 |
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