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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 2726 | . . . 4 β’ (Homf βπΆ) = (Homf βπΆ) | |
4 | 2, 3 | subcssc 17796 | . . 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 17775 | . 2 β’ (π β (ππ½π) β (π(Homf βπΆ)π)) |
8 | eqid 2726 | . . 3 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | subcss2.h | . . 3 β’ π» = (Hom βπΆ) | |
10 | 2, 1, 8 | subcss1 17798 | . . . 4 β’ (π β π β (BaseβπΆ)) |
11 | 10, 5 | sseldd 3978 | . . 3 β’ (π β π β (BaseβπΆ)) |
12 | 10, 6 | sseldd 3978 | . . 3 β’ (π β π β (BaseβπΆ)) |
13 | 3, 8, 9, 11, 12 | homfval 17642 | . 2 β’ (π β (π(Homf βπΆ)π) = (ππ»π)) |
14 | 7, 13 | sseqtrd 4017 | 1 β’ (π β (ππ½π) β (ππ»π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 Γ cxp 5667 Fn wfn 6531 βcfv 6536 (class class class)co 7404 Basecbs 17150 Hom chom 17214 Homf chomf 17616 Subcatcsubc 17762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-pm 8822 df-ixp 8891 df-homf 17620 df-ssc 17763 df-subc 17765 |
This theorem is referenced by: subccatid 17802 funcres 17852 funcres2b 17853 subthinc 47916 |
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