Theorem hmeofn 14481

Description: The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
hmeofn	⊢ Homeo Fn (Top × Top)

Proof of Theorem hmeofn

Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnovex 14375	. . . 4 ⊢ ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → (𝑗 Cn 𝑘) ∈ V)
2		rabexg 4173	. . . 4 ⊢ ((𝑗 Cn 𝑘) ∈ V → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V)
3	1, 2	syl 14	. . 3 ⊢ ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V)
4	3	rgen2a 2548	. 2 ⊢ ∀𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
5		df-hmeo 14480	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
6	5	fnmpo 6257	. 2 ⊢ (∀𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V → Homeo Fn (Top × Top))
7	4, 6	ax-mp 5	1 ⊢ Homeo Fn (Top × Top)

Syntax hints:   ∧ wa 104   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   × cxp 4658  ^◡ccnv 4659   Fn wfn 5250  (class class class)co 5919  Topctop 14176   Cn ccn 14364  Homeochmeo 14479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-top 14177  df-topon 14190  df-cn 14367  df-hmeo 14480

This theorem is referenced by: (None)