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Mirrors > Home > MPE Home > Th. List > subcss2 | Structured version Visualization version GIF version |
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcss1.1 | β’ (π β π½ β (SubcatβπΆ)) |
subcss1.2 | β’ (π β π½ Fn (π Γ π)) |
subcss2.h | β’ π» = (Hom βπΆ) |
subcss2.x | β’ (π β π β π) |
subcss2.y | β’ (π β π β π) |
Ref | Expression |
---|---|
subcss2 | β’ (π β (ππ½π) β (ππ»π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcss1.2 | . . 3 β’ (π β π½ Fn (π Γ π)) | |
2 | subcss1.1 | . . . 4 β’ (π β π½ β (SubcatβπΆ)) | |
3 | eqid 2737 | . . . 4 β’ (Homf βπΆ) = (Homf βπΆ) | |
4 | 2, 3 | subcssc 17733 | . . 3 β’ (π β π½ βcat (Homf βπΆ)) |
5 | subcss2.x | . . 3 β’ (π β π β π) | |
6 | subcss2.y | . . 3 β’ (π β π β π) | |
7 | 1, 4, 5, 6 | ssc2 17712 | . 2 β’ (π β (ππ½π) β (π(Homf βπΆ)π)) |
8 | eqid 2737 | . . 3 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | subcss2.h | . . 3 β’ π» = (Hom βπΆ) | |
10 | 2, 1, 8 | subcss1 17735 | . . . 4 β’ (π β π β (BaseβπΆ)) |
11 | 10, 5 | sseldd 3950 | . . 3 β’ (π β π β (BaseβπΆ)) |
12 | 10, 6 | sseldd 3950 | . . 3 β’ (π β π β (BaseβπΆ)) |
13 | 3, 8, 9, 11, 12 | homfval 17579 | . 2 β’ (π β (π(Homf βπΆ)π) = (ππ»π)) |
14 | 7, 13 | sseqtrd 3989 | 1 β’ (π β (ππ½π) β (ππ»π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3915 Γ cxp 5636 Fn wfn 6496 βcfv 6501 (class class class)co 7362 Basecbs 17090 Hom chom 17151 Homf chomf 17553 Subcatcsubc 17699 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-pm 8775 df-ixp 8843 df-homf 17557 df-ssc 17700 df-subc 17702 |
This theorem is referenced by: subccatid 17739 funcres 17789 funcres2b 17790 subthinc 47134 |
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