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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 6663 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | syl6eqr 2873 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
5 | 4 | rneqd 5801 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
6 | 5 | unieqd 4845 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
7 | df-arw 17282 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
8 | 3 | fvexi 6677 | . . . . . 6 ⊢ 𝐻 ∈ V |
9 | 8 | rnex 7610 | . . . . 5 ⊢ ran 𝐻 ∈ V |
10 | 9 | uniex 7460 | . . . 4 ⊢ ∪ ran 𝐻 ∈ V |
11 | 6, 7, 10 | fvmpt 6761 | . . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
12 | 7 | fvmptndm 6791 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅) |
13 | df-homa 17281 | . . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 13 | fvmptndm 6791 | . . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
15 | 3, 14 | syl5eq 2867 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
16 | 15 | rneqd 5801 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
17 | rn0 5789 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
18 | 16, 17 | syl6eq 2871 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
19 | 18 | unieqd 4845 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
20 | uni0 4859 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
21 | 19, 20 | syl6eq 2871 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
22 | 12, 21 | eqtr4d 2858 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
23 | 11, 22 | pm2.61i 184 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
24 | 1, 23 | eqtri 2843 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 ∅c0 4284 {csn 4560 ∪ cuni 4831 ↦ cmpt 5139 × cxp 5546 ran crn 5549 ‘cfv 6348 Basecbs 16478 Hom chom 16571 Catccat 16930 Arrowcarw 17277 Homachoma 17278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 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 17281 df-arw 17282 |
This theorem is referenced by: arwhoma 17300 homarw 17301 |
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