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Mirrors > Home > MPE Home > Th. List > homaf | Structured version Visualization version GIF version |
Description: Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homarcl.h | β’ π» = (HomaβπΆ) |
homafval.b | β’ π΅ = (BaseβπΆ) |
homafval.c | β’ (π β πΆ β Cat) |
Ref | Expression |
---|---|
homaf | β’ (π β π»:(π΅ Γ π΅)βΆπ« ((π΅ Γ π΅) Γ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homarcl.h | . . 3 β’ π» = (HomaβπΆ) | |
2 | homafval.b | . . 3 β’ π΅ = (BaseβπΆ) | |
3 | homafval.c | . . 3 β’ (π β πΆ β Cat) | |
4 | eqid 2730 | . . 3 β’ (Hom βπΆ) = (Hom βπΆ) | |
5 | 1, 2, 3, 4 | homafval 17985 | . 2 β’ (π β π» = (π₯ β (π΅ Γ π΅) β¦ ({π₯} Γ ((Hom βπΆ)βπ₯)))) |
6 | snssi 4812 | . . . . 5 β’ (π₯ β (π΅ Γ π΅) β {π₯} β (π΅ Γ π΅)) | |
7 | 6 | adantl 480 | . . . 4 β’ ((π β§ π₯ β (π΅ Γ π΅)) β {π₯} β (π΅ Γ π΅)) |
8 | ssv 4007 | . . . 4 β’ ((Hom βπΆ)βπ₯) β V | |
9 | xpss12 5692 | . . . 4 β’ (({π₯} β (π΅ Γ π΅) β§ ((Hom βπΆ)βπ₯) β V) β ({π₯} Γ ((Hom βπΆ)βπ₯)) β ((π΅ Γ π΅) Γ V)) | |
10 | 7, 8, 9 | sylancl 584 | . . 3 β’ ((π β§ π₯ β (π΅ Γ π΅)) β ({π₯} Γ ((Hom βπΆ)βπ₯)) β ((π΅ Γ π΅) Γ V)) |
11 | vsnex 5430 | . . . . 5 β’ {π₯} β V | |
12 | fvex 6905 | . . . . 5 β’ ((Hom βπΆ)βπ₯) β V | |
13 | 11, 12 | xpex 7744 | . . . 4 β’ ({π₯} Γ ((Hom βπΆ)βπ₯)) β V |
14 | 13 | elpw 4607 | . . 3 β’ (({π₯} Γ ((Hom βπΆ)βπ₯)) β π« ((π΅ Γ π΅) Γ V) β ({π₯} Γ ((Hom βπΆ)βπ₯)) β ((π΅ Γ π΅) Γ V)) |
15 | 10, 14 | sylibr 233 | . 2 β’ ((π β§ π₯ β (π΅ Γ π΅)) β ({π₯} Γ ((Hom βπΆ)βπ₯)) β π« ((π΅ Γ π΅) Γ V)) |
16 | 5, 15 | fmpt3d 7118 | 1 β’ (π β π»:(π΅ Γ π΅)βΆπ« ((π΅ Γ π΅) Γ V)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 Vcvv 3472 β wss 3949 π« cpw 4603 {csn 4629 Γ cxp 5675 βΆwf 6540 βcfv 6544 Basecbs 17150 Hom chom 17214 Catccat 17614 Homachoma 17979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-homa 17982 |
This theorem is referenced by: homarcl2 17991 homarel 17992 arwhoma 18001 |
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