Theorem hmeofval 23883

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.

Step	Hyp	Ref	Expression
1		oveq12 7420	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2		oveq12 7420	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
3	2	ancoms 463	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
4	3	eleq2d 2855	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (◡𝑓 ∈ (𝑘 Cn 𝑗) ↔ ◡𝑓 ∈ (𝐾 Cn 𝐽)))
5	1, 4	rabeqbidv 3441	. . 3 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
6		df-hmeo 23880	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
7		ovex 7444	. . . 4 ⊢ (𝐽 Cn 𝐾) ∈ V
8	7	rabex 5310	. . 3 ⊢ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V
9	5, 6, 8	ovmpoa 7566	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
10	6	mpondm0 7651	. . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11		cntop1 23365	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12		cntop2 23366	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
13	11, 12	jca 520	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
14	13	a1d 26	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (◡𝑓 ∈ (𝐾 Cn 𝐽) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)))
15	14	con3rr3 156	. . . . 5 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽)))
16	15	ralrimiv 3162	. . . 4 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽))
17		rabeq0 4352	. . . 4 ⊢ ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽))
18	16, 17	sylibr 237	. . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
19	10, 18	eqtr4d 2807	. 2 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
20	9, 19	pm2.61i 184	1 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  ∅c0 4294  ^◡ccnv 5661  (class class class)co 7411  Topctop 23018   Cn ccn 23349  Homeochmeo 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8825  df-top 23019  df-topon 23036  df-cn 23352  df-hmeo 23880

This theorem is referenced by:  ishmeo  23884