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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 | ⊢ 𝐴 = (Arrow‘𝐶) |
Ref | Expression |
---|---|
arwrcl | ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-arw 17279 | . . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
2 | 1 | dmmptss 6062 | . 2 ⊢ dom Arrow ⊆ Cat |
3 | elfvdm 6677 | . . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow) | |
4 | arwrcl.a | . . 3 ⊢ 𝐴 = (Arrow‘𝐶) | |
5 | 3, 4 | eleq2s 2908 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow) |
6 | 2, 5 | sseldi 3913 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 dom cdm 5519 ran crn 5520 ‘cfv 6324 Catccat 16927 Arrowcarw 17274 Homachoma 17275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 df-arw 17279 |
This theorem is referenced by: arwhoma 17297 coafval 17316 |
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