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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 17399 | . . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
2 | 1 | dmmptss 6073 | . 2 ⊢ dom Arrow ⊆ Cat |
3 | elfvdm 6706 | . . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow) | |
4 | arwrcl.a | . . 3 ⊢ 𝐴 = (Arrow‘𝐶) | |
5 | 3, 4 | eleq2s 2851 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow) |
6 | 2, 5 | sseldi 3875 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∪ cuni 4796 dom cdm 5525 ran crn 5526 ‘cfv 6339 Catccat 17038 Arrowcarw 17394 Homachoma 17395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5531 df-rel 5532 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fv 6347 df-arw 17399 |
This theorem is referenced by: arwhoma 17417 coafval 17436 |
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