Theorem indishmph 22403

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 8501	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 6590	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6594	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
4		vex 3444	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7598	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2899	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
7		f1ofo 6597	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
8		fornex 7639	. . . . . . . . 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 8403	. . . . . . 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 21606	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 21897	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 587	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2893	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6602	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
18		f1of 6590	. . . . . . . 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 8403	. . . . . . 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 21606	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 17	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 21897	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 587	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2893	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 22364	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 586	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 22382	. . . 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 220	1 ⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  {cpr 4527   class class class wbr 5030  ^◡ccnv 5518  dom cdm 5519  ⟶wf 6320  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389   ≈ cen 8489  TopOnctopon 21515   Cn ccn 21829  Homeochmeo 22358   ≃ chmph 22359

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-1o 8085  df-map 8391  df-en 8493  df-top 21499  df-topon 21516  df-cn 21832  df-hmeo 22360  df-hmph 22361

This theorem is referenced by: (None)