Theorem hmphdis 23690

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 4912	. . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽
2		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	pweqi 4582	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽
4	1, 3	sseqtrri 3999	. . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋)
6		hmph 23670	. . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7		n0 4319	. . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8		elpwi 4573	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
9		imassrn 6045	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
10		unipw 5413	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	10	eqcomi 2739	. . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴
12	2, 11	hmeof1o 23658	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴)
13		f1of 6803	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴)
14		frn 6698	. . . . . . . . . . . . 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 3961	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴)
18		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
19	18	imaex 7893	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V
20	19	elpw 4570	. . . . . . . . . 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 23660	. . . . . . . . 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 3955	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
27	26	exlimiv 1930	. . . 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 3967	1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874   class class class wbr 5110  ran crn 5642   “ cima 5644  ⟶wf 6510  –1-1-onto→wf1o 6513  (class class class)co 7390  Homeochmeo 23647   ≃ chmph 23648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-map 8804  df-top 22788  df-topon 22805  df-cn 23121  df-hmeo 23649  df-hmph 23650

This theorem is referenced by: (None)