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| Mirrors > Home > MPE Home > Th. List > xkobval | Structured version Visualization version GIF version | ||
| Description: Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| xkoval.x | ⊢ 𝑋 = ∪ 𝑅 | 
| xkoval.k | ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} | 
| xkoval.t | ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | 
| Ref | Expression | 
|---|---|
| xkobval | ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xkoval.t | . . 3 ⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | |
| 2 | 1 | rnmpo 7567 | . 2 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} | 
| 3 | oveq2 7440 | . . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑅 ↾t 𝑥) = (𝑅 ↾t 𝑘)) | |
| 4 | 3 | eleq1d 2825 | . . . . 5 ⊢ (𝑥 = 𝑘 → ((𝑅 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑘) ∈ Comp)) | 
| 5 | 4 | rexrab 3701 | . . . 4 ⊢ (∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) | 
| 6 | xkoval.k | . . . . 5 ⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} | |
| 7 | 6 | rexeqi 3324 | . . . 4 ⊢ (∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | 
| 8 | r19.42v 3190 | . . . . 5 ⊢ (∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) | |
| 9 | 8 | rexbii 3093 | . . . 4 ⊢ (∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 𝑋((𝑅 ↾t 𝑘) ∈ Comp ∧ ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) | 
| 10 | 5, 7, 9 | 3bitr4i 303 | . . 3 ⊢ (∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) | 
| 11 | 10 | abbii 2808 | . 2 ⊢ {𝑠 ∣ ∃𝑘 ∈ 𝐾 ∃𝑣 ∈ 𝑆 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})} | 
| 12 | 2, 11 | eqtri 2764 | 1 ⊢ ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋∃𝑣 ∈ 𝑆 ((𝑅 ↾t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {crab 3435 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 ran crn 5685 “ cima 5687 (class class class)co 7432 ∈ cmpo 7434 ↾t crest 17466 Cn ccn 23233 Compccmp 23395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 | 
| This theorem is referenced by: xkoccn 23628 xkoco1cn 23666 xkoco2cn 23667 xkoinjcn 23696 | 
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