Theorem indishmph 22406

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 8518	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 6615	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6619	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
4		vex 3497	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7616	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2922	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
7		f1ofo 6622	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
8		fornex 7657	. . . . . . . . 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 8420	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
11	2, 10	mpbird 259	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ (𝐵 ↑m 𝐴))
12		indistopon 21609	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 21900	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 586	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2916	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6627	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
18		f1of 6615	. . . . . . . 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 8420	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (◡𝑓 ∈ (𝐴 ↑m 𝐵) ↔ ◡𝑓:𝐵⟶𝐴))
21	19, 20	mpbird 259	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ (𝐴 ↑m 𝐵))
22		indistopon 21609	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 21900	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 586	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2916	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 22367	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 585	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 22385	. . . 4 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
30	28, 29	syl 17	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
31	30	exlimiv 1931	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
32	1, 31	sylbi 219	1 ⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  {cpr 4569   class class class wbr 5066  ^◡ccnv 5554  dom cdm 5555  ⟶wf 6351  –onto→wfo 6353  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406   ≈ cen 8506  TopOnctopon 21518   Cn ccn 21832  Homeochmeo 22361   ≃ chmph 22362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-1o 8102  df-map 8408  df-en 8510  df-top 21502  df-topon 21519  df-cn 21835  df-hmeo 22363  df-hmph 22364

This theorem is referenced by: (None)