Theorem hmphdis 23786

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 4946	. . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽
2		hmphdis.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	pweqi 4614	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽
4	1, 3	sseqtrri 4017	. . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋)
6		hmph 23766	. . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7		n0 4347	. . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8		elpwi 4605	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
9		imassrn 6071	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
10		unipw 5447	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	10	eqcomi 2735	. . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴
12	2, 11	hmeof1o 23754	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴)
13		f1of 6833	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴)
14		frn 6725	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴)
15	12, 13, 14	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴)
16	15	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴)
17	9, 16	sstrid 3991	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴)
18		vex 3467	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
19	18	imaex 7917	. . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V
20	19	elpw 4602	. . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴)
21	17, 20	sylibr 233	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴)
22	2	hmeoopn 23756	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴))
23	21, 22	mpbird 256	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽)
24	23	ex 411	. . . . . . 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 3985	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
27	26	exlimiv 1926	. . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽)
28	7, 27	sylbi 216	. . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽)
29	6, 28	sylbi 216	. 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽)
30	5, 29	eqssd 3997	1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930   ⊆ wss 3947  ∅c0 4323  𝒫 cpw 4598  ∪ cuni 4906   class class class wbr 5144  ran crn 5674   “ cima 5676  ⟶wf 6540  –1-1-onto→wf1o 6543  (class class class)co 7414  Homeochmeo 23743   ≃ chmph 23744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7993  df-2nd 7994  df-1o 8486  df-map 8847  df-top 22882  df-topon 22899  df-cn 23217  df-hmeo 23745  df-hmph 23746

This theorem is referenced by: (None)