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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 6906 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
| 3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
| 5 | 4 | rneqd 5949 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
| 6 | 5 | unieqd 4920 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
| 7 | df-arw 18072 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
| 8 | 3 | fvexi 6920 | . . . . . 6 ⊢ 𝐻 ∈ V |
| 9 | 8 | rnex 7932 | . . . . 5 ⊢ ran 𝐻 ∈ V |
| 10 | 9 | uniex 7761 | . . . 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 18071 | . . . . . . . . . 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 2789 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
| 16 | 15 | rneqd 5949 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
| 17 | rn0 5936 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
| 18 | 16, 17 | eqtrdi 2793 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
| 19 | 18 | unieqd 4920 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
| 20 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 21 | 19, 20 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
| 22 | 12, 21 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
| 23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
| 24 | 1, 23 | eqtri 2765 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∅c0 4333 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 × cxp 5683 ran crn 5686 ‘cfv 6561 Basecbs 17247 Hom chom 17308 Catccat 17707 Arrowcarw 18067 Homachoma 18068 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-homa 18071 df-arw 18072 |
| This theorem is referenced by: arwhoma 18090 homarw 18091 |
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