Theorem indishmph 24172

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 9021	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
2		f1of 6854	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶𝐵)
3		f1odm 6858	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → dom 𝑓 = 𝐴)
4		vex 3458	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
5	4	dmex 7955	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
6	3, 5	eqeltrrdi 2873	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝐴 ∈ V)
7		f1ofo 6862	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶onto𝐵)
8		focdmex 8002	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑓:𝐴⟶onto𝐵 → 𝐵 ∈ V))
9	6, 7, 8	sylc 66	. . . . . . . 8 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝐵 ∈ V)
10	9, 6	elmapd 8898	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵))
11	2, 10	mpbird 260	. . . . . 6 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓 ∈ (𝐵 ↑m 𝐴))
12		indistopon 23374	. . . . . . . 8 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
13	6, 12	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14		cnindis 23665	. . . . . . 7 ⊢ (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
15	13, 9, 14	syl2anc 596	. . . . . 6 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵 ↑m 𝐴))
16	11, 15	eleqtrrd 2867	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17		f1ocnv 6867	. . . . . . . 8 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓:𝐵⟶1-1onto𝐴)
18		f1of 6854	. . . . . . . 8 ⊢ (◡𝑓:𝐵⟶1-1onto𝐴 → ◡𝑓:𝐵⟶𝐴)
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓:𝐵⟶𝐴)
20	6, 9	elmapd 8898	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → (◡𝑓 ∈ (𝐴 ↑m 𝐵) ↔ ◡𝑓:𝐵⟶𝐴))
21	19, 20	mpbird 260	. . . . . 6 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓 ∈ (𝐴 ↑m 𝐵))
22		indistopon 23374	. . . . . . . 8 ⊢ (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
23	9, 22	syl 18	. . . . . . 7 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24		cnindis 23665	. . . . . . 7 ⊢ (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
25	23, 6, 24	syl2anc 596	. . . . . 6 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴 ↑m 𝐵))
26	21, 25	eleqtrrd 2867	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27		ishmeo 24133	. . . . 5 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ ◡𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
28	16, 26, 27	sylanbrc 595	. . . 4 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29		hmphi 24151	. . . 4 ⊢ (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
30	28, 29	syl 18	. . 3 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
31	30	exlimiv 1965	. 2 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto𝐵 → {∅, 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 2147  Vcvv 3454  ∅c0 4281  {cpr 4590   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658  ⟶wf 6562  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885   ≈ cen 9008  TopOnctopon 23283   Cn ccn 23597  Homeochmeo 24127   ≃ chmph 24128

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-1o 8514  df-map 8887  df-en 9012  df-top 23267  df-topon 23284  df-cn 23600  df-hmeo 24129  df-hmph 24130

This theorem is used by: (None)