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Theorem cmphaushmeo 21543
Description: A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)
Hypotheses
Ref Expression
cmphaushmeo.1 𝑋 = 𝐽
cmphaushmeo.2 𝑌 = 𝐾
Assertion
Ref Expression
cmphaushmeo ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋1-1-onto𝑌))

Proof of Theorem cmphaushmeo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cmphaushmeo.1 . . 3 𝑋 = 𝐽
2 cmphaushmeo.2 . . 3 𝑌 = 𝐾
31, 2hmeof1o 21507 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌)
4 f1ocnv 6116 . . . . . . . 8 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
5 f1of 6104 . . . . . . . 8 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
64, 5syl 17 . . . . . . 7 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋)
76a1i 11 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌𝐹:𝑌𝑋))
8 f1orel 6107 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌 → Rel 𝐹)
98ad2antll 764 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → Rel 𝐹)
10 dfrel2 5552 . . . . . . . . . . 11 (Rel 𝐹𝐹 = 𝐹)
119, 10sylib 208 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹 = 𝐹)
1211imaeq1d 5434 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) = (𝐹𝑥))
13 simp2 1060 . . . . . . . . . . 11 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
1413adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐾 ∈ Haus)
15 imassrn 5446 . . . . . . . . . . 11 (𝐹𝑥) ⊆ ran 𝐹
16 f1ofo 6111 . . . . . . . . . . . . 13 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋onto𝑌)
1716ad2antll 764 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹:𝑋onto𝑌)
18 forn 6085 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌 → ran 𝐹 = 𝑌)
1917, 18syl 17 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → ran 𝐹 = 𝑌)
2015, 19syl5sseq 3638 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ⊆ 𝑌)
21 simpl3 1064 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22 simp1 1059 . . . . . . . . . . . . 13 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2322adantr 481 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐽 ∈ Comp)
24 simprl 793 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝑥 ∈ (Clsd‘𝐽))
25 cmpcld 21145 . . . . . . . . . . . 12 ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽t 𝑥) ∈ Comp)
2623, 24, 25syl2anc 692 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐽t 𝑥) ∈ Comp)
27 imacmp 21140 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑥) ∈ Comp) → (𝐾t (𝐹𝑥)) ∈ Comp)
2821, 26, 27syl2anc 692 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐾t (𝐹𝑥)) ∈ Comp)
292hauscmp 21150 . . . . . . . . . 10 ((𝐾 ∈ Haus ∧ (𝐹𝑥) ⊆ 𝑌 ∧ (𝐾t (𝐹𝑥)) ∈ Comp) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3014, 20, 28, 29syl3anc 1323 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3112, 30eqeltrd 2698 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3231expr 642 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹𝑥) ∈ (Clsd‘𝐾)))
3332ralrimdva 2965 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾)))
347, 33jcad 555 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
35 haustop 21075 . . . . . . . 8 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3613, 35syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
372toptopon 20662 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3836, 37sylib 208 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
39 cmptop 21138 . . . . . . . 8 (𝐽 ∈ Comp → 𝐽 ∈ Top)
4022, 39syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
411toptopon 20662 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4240, 41sylib 208 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
43 iscncl 21013 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐹 ∈ (𝐾 Cn 𝐽) ↔ (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
4438, 42, 43syl2anc 692 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐾 Cn 𝐽) ↔ (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
4534, 44sylibrd 249 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝐾 Cn 𝐽)))
46 simp3 1061 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
4745, 46jctild 565 . . 3 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽))))
48 ishmeo 21502 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
4947, 48syl6ibr 242 . 2 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝐽Homeo𝐾)))
503, 49impbid2 216 1 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋1-1-onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560   cuni 4409  ccnv 5083  ran crn 5085  cima 5087  Rel wrel 5089  wf 5853  ontowfo 5855  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  t crest 16021  Topctop 20638  TopOnctopon 20655  Clsdccld 20760   Cn ccn 20968  Hauscha 21052  Compccmp 21129  Homeochmeo 21496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-fin 7919  df-fi 8277  df-rest 16023  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-cld 20763  df-cls 20765  df-cn 20971  df-haus 21059  df-cmp 21130  df-hmeo 21498
This theorem is referenced by:  cncfcnvcn  22664
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