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Mirrors > Home > MPE Home > Th. List > 0ssc | Structured version Visualization version GIF version |
Description: For any category πΆ, the empty set is a subcategory subset of πΆ. (Contributed by AV, 23-Apr-2020.) |
Ref | Expression |
---|---|
0ssc | β’ (πΆ β Cat β β βcat (Homf βπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4392 | . . 3 β’ β β (BaseβπΆ) | |
2 | 1 | a1i 11 | . 2 β’ (πΆ β Cat β β β (BaseβπΆ)) |
3 | ral0 4508 | . . 3 β’ βπ₯ β β βπ¦ β β (π₯β π¦) β (π₯(Homf βπΆ)π¦) | |
4 | 3 | a1i 11 | . 2 β’ (πΆ β Cat β βπ₯ β β βπ¦ β β (π₯β π¦) β (π₯(Homf βπΆ)π¦)) |
5 | f0 6772 | . . . . . 6 β’ β :β βΆβ | |
6 | ffn 6716 | . . . . . 6 β’ (β :β βΆβ β β Fn β ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 β’ β Fn β |
8 | xp0 6156 | . . . . . 6 β’ (β Γ β ) = β | |
9 | 8 | fneq2i 6646 | . . . . 5 β’ (β Fn (β Γ β ) β β Fn β ) |
10 | 7, 9 | mpbir 230 | . . . 4 β’ β Fn (β Γ β ) |
11 | 10 | a1i 11 | . . 3 β’ (πΆ β Cat β β Fn (β Γ β )) |
12 | eqid 2728 | . . . . 5 β’ (Homf βπΆ) = (Homf βπΆ) | |
13 | eqid 2728 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
14 | 12, 13 | homffn 17666 | . . . 4 β’ (Homf βπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ)) |
15 | 14 | a1i 11 | . . 3 β’ (πΆ β Cat β (Homf βπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ))) |
16 | fvexd 6906 | . . 3 β’ (πΆ β Cat β (BaseβπΆ) β V) | |
17 | 11, 15, 16 | isssc 17796 | . 2 β’ (πΆ β Cat β (β βcat (Homf βπΆ) β (β β (BaseβπΆ) β§ βπ₯ β β βπ¦ β β (π₯β π¦) β (π₯(Homf βπΆ)π¦)))) |
18 | 2, 4, 17 | mpbir2and 712 | 1 β’ (πΆ β Cat β β βcat (Homf βπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2099 βwral 3057 Vcvv 3470 β wss 3945 β c0 4318 class class class wbr 5142 Γ cxp 5670 Fn wfn 6537 βΆwf 6538 βcfv 6542 (class class class)co 7414 Basecbs 17173 Catccat 17637 Homf chomf 17639 βcat cssc 17783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-ixp 8910 df-homf 17643 df-ssc 17786 |
This theorem is referenced by: 0subcat 17817 |
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