Theorem hmphdis 24145

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 4909	. . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽
2		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	pweqi 4576	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽
4	1, 3	sseqtrri 3962	. . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋)
6		hmph 24125	. . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7		n0 4303	. . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8		elpwi 4567	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
9		imassrn 6210	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
10		unipw 5422	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	10	eqcomi 2772	. . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴
12	2, 11	hmeof1o 24113	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋⟶1-1onto𝐴)
13		f1of 6846	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶1-1onto𝐴 → 𝑓:𝑋⟶𝐴)
14		frn 6739	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴)
15	12, 13, 14	3syl 19	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴)
16	15	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴)
17	9, 16	sstrid 3924	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴)
18		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
19	18	imaex 7950	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V
20	19	elpw 4564	. . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴)
21	17, 20	sylibr 237	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴)
22	2	hmeoopn 24115	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴))
23	21, 22	mpbird 260	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽)
24	23	ex 418	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐽))
25	8, 24	syl5 35	. . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋 → 𝑥 ∈ 𝐽))
26	25	ssrdv 3919	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
27	26	exlimiv 1965	. . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
28	7, 27	sylbi 220	. . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽)
29	6, 28	sylbi 220	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽)
30	5, 29	eqssd 3930	1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  ran crn 5656   “ cima 5658  ⟶wf 6554  ⟶1-1ontowf1o 6557  (class class class)co 7443  Homeochmeo 24102   ≃ chmph 24103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-1o 8504  df-map 8877  df-top 23242  df-topon 23259  df-cn 23575  df-hmeo 24104  df-hmph 24105

This theorem is used by: (None)