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Mirrors > Home > MPE Home > Th. List > homarcl2 | Structured version Visualization version GIF version |
Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | β’ π» = (HomaβπΆ) |
homarcl2.b | β’ π΅ = (BaseβπΆ) |
Ref | Expression |
---|---|
homarcl2 | β’ (πΉ β (ππ»π) β (π β π΅ β§ π β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6939 | . . . 4 β’ (πΉ β (π»ββ¨π, πβ©) β β¨π, πβ© β dom π») | |
2 | df-ov 7429 | . . . 4 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
3 | 1, 2 | eleq2s 2847 | . . 3 β’ (πΉ β (ππ»π) β β¨π, πβ© β dom π») |
4 | homahom.h | . . . . 5 β’ π» = (HomaβπΆ) | |
5 | homarcl2.b | . . . . 5 β’ π΅ = (BaseβπΆ) | |
6 | 4 | homarcl 18026 | . . . . 5 β’ (πΉ β (ππ»π) β πΆ β Cat) |
7 | 4, 5, 6 | homaf 18028 | . . . 4 β’ (πΉ β (ππ»π) β π»:(π΅ Γ π΅)βΆπ« ((π΅ Γ π΅) Γ V)) |
8 | 7 | fdmd 6738 | . . 3 β’ (πΉ β (ππ»π) β dom π» = (π΅ Γ π΅)) |
9 | 3, 8 | eleqtrd 2831 | . 2 β’ (πΉ β (ππ»π) β β¨π, πβ© β (π΅ Γ π΅)) |
10 | opelxp 5718 | . 2 β’ (β¨π, πβ© β (π΅ Γ π΅) β (π β π΅ β§ π β π΅)) | |
11 | 9, 10 | sylib 217 | 1 β’ (πΉ β (ππ»π) β (π β π΅ β§ π β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 π« cpw 4606 β¨cop 4638 Γ cxp 5680 dom cdm 5682 βcfv 6553 (class class class)co 7426 Basecbs 17189 Homachoma 18021 |
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 7748 |
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-homa 18024 |
This theorem is referenced by: homarel 18034 homa1 18035 homahom2 18036 homadm 18038 homacd 18039 arwdm 18045 arwcd 18046 coahom 18068 arwlid 18070 arwrid 18071 arwass 18072 |
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