Theorem indishmph 23788

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 6774	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6778	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
4		vex 3436	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7856	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2849	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
7		f1ofo 6781	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
8		focdmex 7905	. . . . . . . . 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 8784	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
11	2, 10	mpbird 258	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ (𝐵 ↑m 𝐴))
12		indistopon 22991	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 23282	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 590	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2843	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6786	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
18		f1of 6774	. . . . . . . 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 8784	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡𝑓 ∈ (𝐴 ↑m 𝐵) ↔ ◡𝑓:𝐵⟶𝐴))
21	19, 20	mpbird 258	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ (𝐴 ↑m 𝐵))
22		indistopon 22991	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 23282	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 590	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2843	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 23749	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 589	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 23767	. . . 4 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
30	28, 29	syl 17	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
31	30	exlimiv 1937	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
32	1, 31	sylbi 218	1 ⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Syntax hints:   → wi 4   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  {cpr 4564   class class class wbr 5079  ^◡ccnv 5624  dom cdm 5625  ⟶wf 6488  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770   ≈ cen 8887  TopOnctopon 22900   Cn ccn 23214  Homeochmeo 23743   ≃ chmph 23744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-1o 8402  df-map 8772  df-en 8891  df-top 22884  df-topon 22901  df-cn 23217  df-hmeo 23745  df-hmph 23746

This theorem is referenced by: (None)