Theorem hmeofn 12476

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 12370	. . . 4 ⊢ ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → (𝑗 Cn 𝑘) ∈ V)
2		rabexg 4071	. . . 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 2486	. 2 ⊢ ∀𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
5		df-hmeo 12475	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
6	5	fnmpo 6100	. 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 103   ∈ wcel 1480  ∀wral 2416  {crab 2420  Vcvv 2686   × cxp 4537  ^◡ccnv 4538   Fn wfn 5118  (class class class)co 5774  Topctop 12169   Cn ccn 12359  Homeochmeo 12474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12170  df-topon 12183  df-cn 12362  df-hmeo 12475

This theorem is referenced by: (None)