Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnfex | Structured version Visualization version GIF version |
Description: The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
cnfex | ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | jctr 525 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽)) |
3 | istopon 21448 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ 𝐽)) | |
4 | 2, 3 | sylibr 235 | . . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | eqid 2818 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
6 | 5 | jctr 525 | . . . 4 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾)) |
7 | istopon 21448 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐾)) | |
8 | 6, 7 | sylibr 235 | . . 3 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
9 | cnfval 21769 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}) | |
10 | 4, 8, 9 | syl2an 595 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) = {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}) |
11 | uniexg 7456 | . . . . 5 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V) | |
12 | uniexg 7456 | . . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V) | |
13 | mapvalg 8405 | . . . . 5 ⊢ ((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾}) | |
14 | 11, 12, 13 | syl2anr 596 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) = {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾}) |
15 | mapex 8401 | . . . . 5 ⊢ ((∪ 𝐽 ∈ V ∧ ∪ 𝐾 ∈ V) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V) | |
16 | 12, 11, 15 | syl2an 595 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∣ 𝑓:∪ 𝐽⟶∪ 𝐾} ∈ V) |
17 | 14, 16 | eqeltrd 2910 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (∪ 𝐾 ↑m ∪ 𝐽) ∈ V) |
18 | rabexg 5225 | . . 3 ⊢ ((∪ 𝐾 ↑m ∪ 𝐽) ∈ V → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V) | |
19 | 17, 18 | syl 17 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (∪ 𝐾 ↑m ∪ 𝐽) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ∈ V) |
20 | 10, 19 | eqeltrd 2910 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 {crab 3139 Vcvv 3492 ∪ cuni 4830 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 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-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-topon 21447 df-cn 21763 |
This theorem is referenced by: stoweidlem53 42215 stoweidlem57 42219 |
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