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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 6878 | . . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶)) | |
3 | arwval.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
4 | 2, 3 | eqtr4di 2789 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻) |
5 | 4 | rneqd 5929 | . . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻) |
6 | 5 | unieqd 4915 | . . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻) |
7 | df-arw 17959 | . . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) | |
8 | 3 | fvexi 6892 | . . . . . 6 ⊢ 𝐻 ∈ V |
9 | 8 | rnex 7885 | . . . . 5 ⊢ ran 𝐻 ∈ V |
10 | 9 | uniex 7714 | . . . 4 ⊢ ∪ ran 𝐻 ∈ V |
11 | 6, 7, 10 | fvmpt 6984 | . . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
12 | 7 | fvmptndm 7014 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅) |
13 | df-homa 17958 | . . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) | |
14 | 13 | fvmptndm 7014 | . . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅) |
15 | 3, 14 | eqtrid 2783 | . . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅) |
16 | 15 | rneqd 5929 | . . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅) |
17 | rn0 5917 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
18 | 16, 17 | eqtrdi 2787 | . . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅) |
19 | 18 | unieqd 4915 | . . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅) |
20 | uni0 4932 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
21 | 19, 20 | eqtrdi 2787 | . . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅) |
22 | 12, 21 | eqtr4d 2774 | . . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻) |
23 | 11, 22 | pm2.61i 182 | . 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻 |
24 | 1, 23 | eqtri 2759 | 1 ⊢ 𝐴 = ∪ ran 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 ∅c0 4318 {csn 4622 ∪ cuni 4901 ↦ cmpt 5224 × cxp 5667 ran crn 5670 ‘cfv 6532 Basecbs 17126 Hom chom 17190 Catccat 17590 Arrowcarw 17954 Homachoma 17955 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fv 6540 df-homa 17958 df-arw 17959 |
This theorem is referenced by: arwhoma 17977 homarw 17978 |
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