Theorem hmeofn 13673

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 13567	. . . 4 ⊢ ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → (𝑗 Cn 𝑘) ∈ V)
2		rabexg 4145	. . . 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 2531	. 2 ⊢ ∀𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
5		df-hmeo 13672	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
6	5	fnmpo 6200	. 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 2148  ∀wral 2455  {crab 2459  Vcvv 2737   × cxp 4623  ^◡ccnv 4624   Fn wfn 5210  (class class class)co 5872  Topctop 13366   Cn ccn 13556  Homeochmeo 13671

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-map 6647  df-top 13367  df-topon 13380  df-cn 13559  df-hmeo 13672

This theorem is referenced by: (None)