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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 6922 | . . . 4 β’ (πΉ β (π»ββ¨π, πβ©) β β¨π, πβ© β dom π») | |
2 | df-ov 7408 | . . . 4 β’ (ππ»π) = (π»ββ¨π, πβ©) | |
3 | 1, 2 | eleq2s 2845 | . . 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 17990 | . . . . 5 β’ (πΉ β (ππ»π) β πΆ β Cat) |
7 | 4, 5, 6 | homaf 17992 | . . . 4 β’ (πΉ β (ππ»π) β π»:(π΅ Γ π΅)βΆπ« ((π΅ Γ π΅) Γ V)) |
8 | 7 | fdmd 6722 | . . 3 β’ (πΉ β (ππ»π) β dom π» = (π΅ Γ π΅)) |
9 | 3, 8 | eleqtrd 2829 | . 2 β’ (πΉ β (ππ»π) β β¨π, πβ© β (π΅ Γ π΅)) |
10 | opelxp 5705 | . 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 π« cpw 4597 β¨cop 4629 Γ cxp 5667 dom cdm 5669 βcfv 6537 (class class class)co 7405 Basecbs 17153 Homachoma 17985 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-homa 17988 |
This theorem is referenced by: homarel 17998 homa1 17999 homahom2 18000 homadm 18002 homacd 18003 arwdm 18009 arwcd 18010 coahom 18032 arwlid 18034 arwrid 18035 arwass 18036 |
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