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Mirrors > Home > MPE Home > Th. List > hmeofval | Structured version Visualization version GIF version |
Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeofval | ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7440 | . . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾)) | |
2 | oveq12 7440 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽)) | |
3 | 2 | ancoms 458 | . . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽)) |
4 | 3 | eleq2d 2825 | . . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (◡𝑓 ∈ (𝑘 Cn 𝑗) ↔ ◡𝑓 ∈ (𝐾 Cn 𝐽))) |
5 | 1, 4 | rabeqbidv 3452 | . . 3 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}) |
6 | df-hmeo 23779 | . . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) | |
7 | ovex 7464 | . . . 4 ⊢ (𝐽 Cn 𝐾) ∈ V | |
8 | 7 | rabex 5345 | . . 3 ⊢ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V |
9 | 5, 6, 8 | ovmpoa 7588 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}) |
10 | 6 | mpondm0 7673 | . . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅) |
11 | cntop1 23264 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
12 | cntop2 23265 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
13 | 11, 12 | jca 511 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
14 | 13 | a1d 25 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (◡𝑓 ∈ (𝐾 Cn 𝐽) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))) |
15 | 14 | con3rr3 155 | . . . . 5 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽))) |
16 | 15 | ralrimiv 3143 | . . . 4 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽)) |
17 | rabeq0 4394 | . . . 4 ⊢ ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽)) | |
18 | 16, 17 | sylibr 234 | . . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅) |
19 | 10, 18 | eqtr4d 2778 | . 2 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}) |
20 | 9, 19 | pm2.61i 182 | 1 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∅c0 4339 ◡ccnv 5688 (class class class)co 7431 Topctop 22915 Cn ccn 23248 Homeochmeo 23777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-top 22916 df-topon 22933 df-cn 23251 df-hmeo 23779 |
This theorem is referenced by: ishmeo 23783 |
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