Theorem hmeofn 13887

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 13781	. . . 4 ⊢ ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → (𝑗 Cn 𝑘) ∈ V)
2		rabexg 4148	. . . 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 13886	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
6	5	fnmpo 6205	. 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 2739   × cxp 4626  ^◡ccnv 4627   Fn wfn 5213  (class class class)co 5877  Topctop 13582   Cn ccn 13770  Homeochmeo 13885

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-top 13583  df-topon 13596  df-cn 13773  df-hmeo 13886

This theorem is referenced by: (None)