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Mirrors > Home > MPE Home > Th. List > hmeofn | Structured version Visualization version GIF version |
Description: The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmeofn | ⊢ Homeo Fn (Top × Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hmeo 22934 | . 2 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) | |
2 | ovex 7328 | . . 3 ⊢ (𝑗 Cn 𝑘) ∈ V | |
3 | 2 | rabex 5259 | . 2 ⊢ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V |
4 | 1, 3 | fnmpoi 7930 | 1 ⊢ Homeo Fn (Top × Top) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2101 {crab 3221 × cxp 5589 ◡ccnv 5590 Fn wfn 6442 (class class class)co 7295 Topctop 22070 Cn ccn 22403 Homeochmeo 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-hmeo 22934 |
This theorem is referenced by: hmph 22955 hmphtop 22957 hmpher 22963 |
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