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Mirrors > Home > MPE Home > Th. List > arwval | Structured version Visualization version GIF version |
Description: The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwval.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwval.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
arwval | ⊢ 𝐴 = ∪ ran 𝐻 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwval.a | . 2 ⊢ 𝐴 = (Arrow‘𝐶) | |
2 | fveq2 6907 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
5 | 4 | rneqd 5952 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
6 | 5 | unieqd 4925 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
7 | df-arw 18081 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
8 | 3 | fvexi 6921 | . . . . . 6 ⊢ 𝐻 ∈ V |
9 | 8 | rnex 7933 | . . . . 5 ⊢ ran 𝐻 ∈ V |
10 | 9 | uniex 7760 | . . . 4 ⊢ ∪ ran 𝐻 ∈ V |
11 | 6, 7, 10 | fvmpt 7016 | . . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
12 | 7 | fvmptndm 7047 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅) |
13 | df-homa 18080 | . . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 13 | fvmptndm 7047 | . . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
15 | 3, 14 | eqtrid 2787 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
16 | 15 | rneqd 5952 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
17 | rn0 5939 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
18 | 16, 17 | eqtrdi 2791 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
19 | 18 | unieqd 4925 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
20 | uni0 4940 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
21 | 19, 20 | eqtrdi 2791 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
22 | 12, 21 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
24 | 1, 23 | eqtri 2763 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {csn 4631 ∪ cuni 4912 ↦ cmpt 5231 × cxp 5687 ran crn 5690 ‘cfv 6563 Basecbs 17245 Hom chom 17309 Catccat 17709 Arrowcarw 18076 Homachoma 18077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-homa 18080 df-arw 18081 |
This theorem is referenced by: arwhoma 18099 homarw 18100 |
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