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Mirrors > Home > MPE Home > Th. List > arwrcl | Structured version Visualization version GIF version |
Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | β’ π΄ = (ArrowβπΆ) |
Ref | Expression |
---|---|
arwrcl | β’ (πΉ β π΄ β πΆ β Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-arw 17985 | . . 3 β’ Arrow = (π β Cat β¦ βͺ ran (Homaβπ)) | |
2 | 1 | dmmptss 6231 | . 2 β’ dom Arrow β Cat |
3 | elfvdm 6919 | . . 3 β’ (πΉ β (ArrowβπΆ) β πΆ β dom Arrow) | |
4 | arwrcl.a | . . 3 β’ π΄ = (ArrowβπΆ) | |
5 | 3, 4 | eleq2s 2843 | . 2 β’ (πΉ β π΄ β πΆ β dom Arrow) |
6 | 2, 5 | sselid 3973 | 1 β’ (πΉ β π΄ β πΆ β Cat) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βͺ cuni 4900 dom cdm 5667 ran crn 5668 βcfv 6534 Catccat 17613 Arrowcarw 17980 Homachoma 17981 |
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-sep 5290 ax-nul 5297 ax-pr 5418 |
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-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fv 6542 df-arw 17985 |
This theorem is referenced by: arwhoma 18003 coafval 18022 |
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