Theorem indishmph 23984

Description: Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
indishmph	⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Proof of Theorem indishmph

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8955	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 6824	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6828	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
4		vex 3461	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7908	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2874	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
7		f1ofo 6832	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
8		focdmex 7955	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ V))
9	6, 7, 8	sylc 66	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
10	9, 6	elmapd 8839	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
11	2, 10	mpbird 260	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ (𝐵 ↑m 𝐴))
12		indistopon 23187	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 23478	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 596	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2868	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6837	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
18		f1of 6824	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵⟶𝐴)
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵⟶𝐴)
20	6, 9	elmapd 8839	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡𝑓 ∈ (𝐴 ↑m 𝐵) ↔ ◡𝑓:𝐵⟶𝐴))
21	19, 20	mpbird 260	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ (𝐴 ↑m 𝐵))
22		indistopon 23187	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 23478	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 596	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2868	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 23945	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 595	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 23963	. . . 4 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
30	28, 29	syl 18	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
31	30	exlimiv 1963	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
32	1, 31	sylbi 220	1 ⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  {cpr 4593   class class class wbr 5111  ^◡ccnv 5662  dom cdm 5663  ⟶wf 6536  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416   ↑_m cmap 8826   ≈ cen 8942  TopOnctopon 23096   Cn ccn 23410  Homeochmeo 23939   ≃ chmph 23940

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-1o 8455  df-map 8828  df-en 8946  df-top 23080  df-topon 23097  df-cn 23413  df-hmeo 23941  df-hmph 23942

This theorem is used by: (None)