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Mirrors > Home > MPE Home > Th. List > 0subcat | Structured version Visualization version GIF version |
Description: For any category πΆ, the empty set is a (full) subcategory of πΆ, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.) |
Ref | Expression |
---|---|
0subcat | β’ (πΆ β Cat β β β (SubcatβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ssc 17830 | . 2 β’ (πΆ β Cat β β βcat (Homf βπΆ)) | |
2 | ral0 4516 | . . 3 β’ βπ₯ β β (((IdβπΆ)βπ₯) β (π₯β π₯) β§ βπ¦ β β βπ§ β β βπ β (π₯β π¦)βπ β (π¦β π§)(π(β¨π₯, π¦β©(compβπΆ)π§)π) β (π₯β π§)) | |
3 | 2 | a1i 11 | . 2 β’ (πΆ β Cat β βπ₯ β β (((IdβπΆ)βπ₯) β (π₯β π₯) β§ βπ¦ β β βπ§ β β βπ β (π₯β π¦)βπ β (π¦β π§)(π(β¨π₯, π¦β©(compβπΆ)π§)π) β (π₯β π§))) |
4 | eqid 2728 | . . 3 β’ (Homf βπΆ) = (Homf βπΆ) | |
5 | eqid 2728 | . . 3 β’ (IdβπΆ) = (IdβπΆ) | |
6 | eqid 2728 | . . 3 β’ (compβπΆ) = (compβπΆ) | |
7 | id 22 | . . 3 β’ (πΆ β Cat β πΆ β Cat) | |
8 | f0 6783 | . . . . . 6 β’ β :β βΆβ | |
9 | ffn 6727 | . . . . . 6 β’ (β :β βΆβ β β Fn β ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 β’ β Fn β |
11 | 0xp 5780 | . . . . . 6 β’ (β Γ β ) = β | |
12 | 11 | fneq2i 6657 | . . . . 5 β’ (β Fn (β Γ β ) β β Fn β ) |
13 | 10, 12 | mpbir 230 | . . . 4 β’ β Fn (β Γ β ) |
14 | 13 | a1i 11 | . . 3 β’ (πΆ β Cat β β Fn (β Γ β )) |
15 | 4, 5, 6, 7, 14 | issubc2 17829 | . 2 β’ (πΆ β Cat β (β β (SubcatβπΆ) β (β βcat (Homf βπΆ) β§ βπ₯ β β (((IdβπΆ)βπ₯) β (π₯β π₯) β§ βπ¦ β β βπ§ β β βπ β (π₯β π¦)βπ β (π¦β π§)(π(β¨π₯, π¦β©(compβπΆ)π§)π) β (π₯β π§))))) |
16 | 1, 3, 15 | mpbir2and 711 | 1 β’ (πΆ β Cat β β β (SubcatβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 βwral 3058 β c0 4326 β¨cop 4638 class class class wbr 5152 Γ cxp 5680 Fn wfn 6548 βΆwf 6549 βcfv 6553 (class class class)co 7426 compcco 17252 Catccat 17651 Idccid 17652 Homf chomf 17653 βcat cssc 17797 Subcatcsubc 17799 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-pm 8854 df-ixp 8923 df-homf 17657 df-ssc 17800 df-subc 17802 |
This theorem is referenced by: (None) |
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