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Theorem cmphaushmeo 22728
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 22692 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto𝑌)
4 f1ocnv 6694 . . . . . . . 8 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
5 f1of 6682 . . . . . . . 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 6685 . . . . . . . . . . . 12 (𝐹:𝑋1-1-onto𝑌 → Rel 𝐹)
98ad2antll 729 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → Rel 𝐹)
10 dfrel2 6069 . . . . . . . . . . 11 (Rel 𝐹𝐹 = 𝐹)
119, 10sylib 221 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹 = 𝐹)
1211imaeq1d 5945 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) = (𝐹𝑥))
13 simp2 1139 . . . . . . . . . . 11 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
1413adantr 484 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐾 ∈ Haus)
15 imassrn 5957 . . . . . . . . . . 11 (𝐹𝑥) ⊆ ran 𝐹
16 f1ofo 6689 . . . . . . . . . . . . 13 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋onto𝑌)
1716ad2antll 729 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹:𝑋onto𝑌)
18 forn 6657 . . . . . . . . . . . 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, 19sseqtrid 3969 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ⊆ 𝑌)
21 simpl3 1195 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22 simp1 1138 . . . . . . . . . . . . 13 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2322adantr 484 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝐽 ∈ Comp)
24 simprl 771 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → 𝑥 ∈ (Clsd‘𝐽))
25 cmpcld 22330 . . . . . . . . . . . 12 ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽t 𝑥) ∈ Comp)
2623, 24, 25syl2anc 587 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐽t 𝑥) ∈ Comp)
27 imacmp 22325 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽t 𝑥) ∈ Comp) → (𝐾t (𝐹𝑥)) ∈ Comp)
2821, 26, 27syl2anc 587 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐾t (𝐹𝑥)) ∈ Comp)
292hauscmp 22335 . . . . . . . . . 10 ((𝐾 ∈ Haus ∧ (𝐹𝑥) ⊆ 𝑌 ∧ (𝐾t (𝐹𝑥)) ∈ Comp) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3014, 20, 28, 29syl3anc 1373 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3112, 30eqeltrd 2840 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋1-1-onto𝑌)) → (𝐹𝑥) ∈ (Clsd‘𝐾))
3231expr 460 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹𝑥) ∈ (Clsd‘𝐾)))
3332ralrimdva 3113 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾)))
347, 33jcad 516 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
35 haustop 22259 . . . . . . . 8 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3613, 35syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
372toptopon 21845 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3836, 37sylib 221 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
39 cmptop 22323 . . . . . . . 8 (𝐽 ∈ Comp → 𝐽 ∈ Top)
4022, 39syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
411toptopon 21845 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4240, 41sylib 221 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
43 iscncl 22197 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐹 ∈ (𝐾 Cn 𝐽) ↔ (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
4438, 42, 43syl2anc 587 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐾 Cn 𝐽) ↔ (𝐹:𝑌𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(𝐹𝑥) ∈ (Clsd‘𝐾))))
4534, 44sylibrd 262 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝐾 Cn 𝐽)))
46 simp3 1140 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
4745, 46jctild 529 . . 3 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌 → (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽))))
48 ishmeo 22687 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
4947, 48syl6ibr 255 . 2 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝐽Homeo𝐾)))
503, 49impbid2 229 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2112  wral 3064  wss 3883   cuni 4835  ccnv 5567  ran crn 5569  cima 5571  Rel wrel 5573  wf 6396  ontowfo 6398  1-1-ontowf1o 6399  cfv 6400  (class class class)co 7234  t crest 16957  Topctop 21821  TopOnctopon 21838  Clsdccld 21944   Cn ccn 22152  Hauscha 22236  Compccmp 22314  Homeochmeo 22681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-int 4876  df-iun 4922  df-iin 4923  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-1st 7782  df-2nd 7783  df-1o 8225  df-er 8414  df-map 8533  df-en 8650  df-dom 8651  df-fin 8653  df-fi 9056  df-rest 16959  df-topgen 16980  df-top 21822  df-topon 21839  df-bases 21874  df-cld 21947  df-cls 21949  df-cn 22155  df-haus 22243  df-cmp 22315  df-hmeo 22683
This theorem is referenced by:  cncfcnvcn  23853
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