Theorem hmeofvalg 15294

Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
hmeofvalg	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})

Distinct variable groups:   𝑓,𝐽   𝑓,𝐾

Proof of Theorem hmeofvalg

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

Step	Hyp	Ref	Expression
1		cnovex 15187	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
2		rabexg 4260	. . 3 ⊢ ((𝐽 Cn 𝐾) ∈ V → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V)
3	1, 2	syl 14	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V)
4		oveq12 6067	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
5		oveq12 6067	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
6	5	ancoms 268	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
7	6	eleq2d 2304	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (◡𝑓 ∈ (𝑘 Cn 𝑗) ↔ ◡𝑓 ∈ (𝐾 Cn 𝐽)))
8	4, 7	rabeqbidv 2810	. . 3 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
9		df-hmeo 15292	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
10	8, 9	ovmpoga 6191	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
11	3, 10	mpd3an3 1375	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {crab 2526  Vcvv 2815  ^◡ccnv 4753  (class class class)co 6058  Topctop 14988   Cn ccn 15176  Homeochmeo 15291

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-top 14989  df-topon 15002  df-cn 15179  df-hmeo 15292

This theorem is referenced by:  ishmeo  15295