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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 6645 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | eqtr4di 2851 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
5 | 4 | rneqd 5772 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
6 | 5 | unieqd 4814 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
7 | df-arw 17279 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
8 | 3 | fvexi 6659 | . . . . . 6 ⊢ 𝐻 ∈ V |
9 | 8 | rnex 7599 | . . . . 5 ⊢ ran 𝐻 ∈ V |
10 | 9 | uniex 7447 | . . . 4 ⊢ ∪ ran 𝐻 ∈ V |
11 | 6, 7, 10 | fvmpt 6745 | . . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
12 | 7 | fvmptndm 6775 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅) |
13 | df-homa 17278 | . . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 13 | fvmptndm 6775 | . . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
15 | 3, 14 | syl5eq 2845 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
16 | 15 | rneqd 5772 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
17 | rn0 5760 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
18 | 16, 17 | eqtrdi 2849 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
19 | 18 | unieqd 4814 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
20 | uni0 4828 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
21 | 19, 20 | eqtrdi 2849 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
22 | 12, 21 | eqtr4d 2836 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
23 | 11, 22 | pm2.61i 185 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
24 | 1, 23 | eqtri 2821 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ∅c0 4243 {csn 4525 ∪ cuni 4800 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ‘cfv 6324 Basecbs 16475 Hom chom 16568 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 ax-un 7441 |
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-sbc 3721 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-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-homa 17278 df-arw 17279 |
This theorem is referenced by: arwhoma 17297 homarw 17298 |
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