Theorem indishmph 23186

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 8900	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 6789	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6793	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
4		vex 3450	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7853	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2841	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
7		f1ofo 6796	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
8		focdmex 7893	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ V))
9	6, 7, 8	sylc 65	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
10	9, 6	elmapd 8786	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
11	2, 10	mpbird 256	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ (𝐵 ↑m 𝐴))
12		indistopon 22388	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 22680	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 584	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2835	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6801	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
18		f1of 6789	. . . . . . . 8 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵⟶𝐴)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵⟶𝐴)
20	6, 9	elmapd 8786	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡𝑓 ∈ (𝐴 ↑m 𝐵) ↔ ◡𝑓:𝐵⟶𝐴))
21	19, 20	mpbird 256	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ (𝐴 ↑m 𝐵))
22		indistopon 22388	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 22680	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 584	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2835	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 23147	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 583	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 23165	. . . 4 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
30	28, 29	syl 17	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
31	30	exlimiv 1933	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
32	1, 31	sylbi 216	1 ⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3446  ∅c0 4287  {cpr 4593   class class class wbr 5110  ^◡ccnv 5637  dom cdm 5638  ⟶wf 6497  –onto→wfo 6499  –1-1-onto→wf1o 6500  ‘cfv 6501  (class class class)co 7362   ↑_m cmap 8772   ≈ cen 8887  TopOnctopon 22296   Cn ccn 22612  Homeochmeo 23141   ≃ chmph 23142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-1o 8417  df-map 8774  df-en 8891  df-top 22280  df-topon 22297  df-cn 22615  df-hmeo 23143  df-hmph 23144

This theorem is referenced by: (None)