Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6663 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
5 | 4 | rneqd 5801 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
6 | 5 | unieqd 4840 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
7 | df-arw 17275 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
8 | 3 | fvexi 6677 | . . . . . 6 ⊢ 𝐻 ∈ V |
9 | 8 | rnex 7606 | . . . . 5 ⊢ ran 𝐻 ∈ V |
10 | 9 | uniex 7454 | . . . 4 ⊢ ∪ ran 𝐻 ∈ V |
11 | 6, 7, 10 | fvmpt 6761 | . . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
12 | 7 | fvmptndm 6790 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅) |
13 | df-homa 17274 | . . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 13 | fvmptndm 6790 | . . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
15 | 3, 14 | syl5eq 2865 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
16 | 15 | rneqd 5801 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
17 | rn0 5789 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
18 | 16, 17 | syl6eq 2869 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
19 | 18 | unieqd 4840 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
20 | uni0 4857 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
21 | 19, 20 | syl6eq 2869 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
22 | 12, 21 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
23 | 11, 22 | pm2.61i 183 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
24 | 1, 23 | eqtri 2841 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 ∅c0 4288 {csn 4557 ∪ cuni 4830 ↦ cmpt 5137 × cxp 5546 ran crn 5549 ‘cfv 6348 Basecbs 16471 Hom chom 16564 Catccat 16923 Arrowcarw 17270 Homachoma 17271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-homa 17274 df-arw 17275 |
This theorem is referenced by: arwhoma 17293 homarw 17294 |
Copyright terms: Public domain | W3C validator |