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Theorem hmeofval 22060
 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
hmeofval (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
Distinct variable groups:   𝑓,𝐽   𝑓,𝐾

Proof of Theorem hmeofval
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6979 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2 oveq12 6979 . . . . . 6 ((𝑘 = 𝐾𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
32ancoms 451 . . . . 5 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
43eleq2d 2845 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑓 ∈ (𝑘 Cn 𝑗) ↔ 𝑓 ∈ (𝐾 Cn 𝐽)))
51, 4rabeqbidv 3402 . . 3 ((𝑗 = 𝐽𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
6 df-hmeo 22057 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
7 ovex 7002 . . . 4 (𝐽 Cn 𝐾) ∈ V
87rabex 5085 . . 3 {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V
95, 6, 8ovmpoa 7115 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
106mpondm0 7199 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11 cntop1 21542 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12 cntop2 21543 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
1311, 12jca 504 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
1413a1d 25 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐽) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)))
1514con3rr3 153 . . . . 5 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ 𝑓 ∈ (𝐾 Cn 𝐽)))
1615ralrimiv 3125 . . . 4 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
17 rabeq0 4219 . . . 4 ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
1816, 17sylibr 226 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
1910, 18eqtr4d 2811 . 2 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
209, 19pm2.61i 177 1 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  {crab 3086  ∅c0 4173  ◡ccnv 5399  (class class class)co 6970  Topctop 21195   Cn ccn 21526  Homeochmeo 22055 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-map 8200  df-top 21196  df-topon 21213  df-cn 21529  df-hmeo 22057 This theorem is referenced by:  ishmeo  22061
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