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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 4389 | . . 3 β’ β β (BaseβπΆ) | |
2 | 1 | a1i 11 | . 2 β’ (πΆ β Cat β β β (BaseβπΆ)) |
3 | ral0 4505 | . . 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 6763 | . . . . . 6 β’ β :β βΆβ | |
6 | ffn 6708 | . . . . . 6 β’ (β :β βΆβ β β Fn β ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 β’ β Fn β |
8 | xp0 6148 | . . . . . 6 β’ (β Γ β ) = β | |
9 | 8 | fneq2i 6638 | . . . . 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 2724 | . . . . 5 β’ (Homf βπΆ) = (Homf βπΆ) | |
13 | eqid 2724 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
14 | 12, 13 | homffn 17638 | . . . 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 6897 | . . 3 β’ (πΆ β Cat β (BaseβπΆ) β V) | |
17 | 11, 15, 16 | isssc 17768 | . 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 710 | 1 β’ (πΆ β Cat β β βcat (Homf βπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 βwral 3053 Vcvv 3466 β wss 3941 β c0 4315 class class class wbr 5139 Γ cxp 5665 Fn wfn 6529 βΆwf 6530 βcfv 6534 (class class class)co 7402 Basecbs 17145 Catccat 17609 Homf chomf 17611 βcat cssc 17755 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-ixp 8889 df-homf 17615 df-ssc 17758 |
This theorem is referenced by: 0subcat 17789 |
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