Theorem hmeofval 23483

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 7421	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2		oveq12 7421	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
3	2	ancoms 458	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
4	3	eleq2d 2818	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → (◡𝑓 ∈ (𝑘 Cn 𝑗) ↔ ◡𝑓 ∈ (𝐾 Cn 𝐽)))
5	1, 4	rabeqbidv 3448	. . 3 ⊢ ((𝑗 = 𝐽 ∧ 𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
6		df-hmeo 23480	. . 3 ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
7		ovex 7445	. . . 4 ⊢ (𝐽 Cn 𝐾) ∈ V
8	7	rabex 5332	. . 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 7650	. . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11		cntop1 22965	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12		cntop2 22966	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
13	11, 12	jca 511	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
14	13	a1d 25	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → (◡𝑓 ∈ (𝐾 Cn 𝐽) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)))
15	14	con3rr3 155	. . . . 5 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽)))
16	15	ralrimiv 3144	. . . 4 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽))
17		rabeq0 4384	. . . 4 ⊢ ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ ◡𝑓 ∈ (𝐾 Cn 𝐽))
18	16, 17	sylibr 233	. . 3 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
19	10, 18	eqtr4d 2774	. 2 ⊢ (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)})
20	9, 19	pm2.61i 182	1 ⊢ (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431  ∅c0 4322  ^◡ccnv 5675  (class class class)co 7412  Topctop 22616   Cn ccn 22949  Homeochmeo 23478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8825  df-top 22617  df-topon 22634  df-cn 22952  df-hmeo 23480

This theorem is referenced by:  ishmeo  23484