Theorem hmphdis 23704

Description: Homeomorphisms preserve topological discreteness. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmphdis.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hmphdis	⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)

Proof of Theorem hmphdis

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwuni 4894	. . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽
2		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	pweqi 4564	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽
4	1, 3	sseqtrri 3982	. . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋)
6		hmph 23684	. . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7		n0 4301	. . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8		elpwi 4555	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
9		imassrn 6017	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
10		unipw 5389	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	10	eqcomi 2739	. . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴
12	2, 11	hmeof1o 23672	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴)
13		f1of 6759	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴)
14		frn 6654	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴)
15	12, 13, 14	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴)
16	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴)
17	9, 16	sstrid 3944	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴)
18		vex 3438	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
19	18	imaex 7839	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V
20	19	elpw 4552	. . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴)
21	17, 20	sylibr 234	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴)
22	2	hmeoopn 23674	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴))
23	21, 22	mpbird 257	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽)
24	23	ex 412	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐽))
25	8, 24	syl5 34	. . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋 → 𝑥 ∈ 𝐽))
26	25	ssrdv 3938	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
27	26	exlimiv 1931	. . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
28	7, 27	sylbi 217	. . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽)
29	6, 28	sylbi 217	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽)
30	5, 29	eqssd 3950	1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2110   ≠ wne 2926   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4548  ∪ cuni 4857   class class class wbr 5089  ran crn 5615   “ cima 5617  ⟶wf 6473  –1-1-onto→wf1o 6476  (class class class)co 7341  Homeochmeo 23661   ≃ chmph 23662

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-1o 8380  df-map 8747  df-top 22802  df-topon 22819  df-cn 23135  df-hmeo 23663  df-hmph 23664

This theorem is referenced by: (None)