![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17728 | . 2 β’ (πΆ β Cat β β βcat (Homf βπΆ)) | |
2 | ral0 4471 | . . 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 2733 | . . 3 β’ (Homf βπΆ) = (Homf βπΆ) | |
5 | eqid 2733 | . . 3 β’ (IdβπΆ) = (IdβπΆ) | |
6 | eqid 2733 | . . 3 β’ (compβπΆ) = (compβπΆ) | |
7 | id 22 | . . 3 β’ (πΆ β Cat β πΆ β Cat) | |
8 | f0 6724 | . . . . . 6 β’ β :β βΆβ | |
9 | ffn 6669 | . . . . . 6 β’ (β :β βΆβ β β Fn β ) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 β’ β Fn β |
11 | 0xp 5731 | . . . . . 6 β’ (β Γ β ) = β | |
12 | 11 | fneq2i 6601 | . . . . 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 17727 | . 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 712 | 1 β’ (πΆ β Cat β β β (SubcatβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 βwral 3061 β c0 4283 β¨cop 4593 class class class wbr 5106 Γ cxp 5632 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 compcco 17150 Catccat 17549 Idccid 17550 Homf chomf 17551 βcat cssc 17695 Subcatcsubc 17697 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-pm 8771 df-ixp 8839 df-homf 17555 df-ssc 17698 df-subc 17700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |