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Theorem cmphaushmeo 23311
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 = βˆͺ 𝐽
cmphaushmeo.2 π‘Œ = βˆͺ 𝐾
Assertion
Ref Expression
cmphaushmeo ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-ontoβ†’π‘Œ))

Proof of Theorem cmphaushmeo
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 cmphaushmeo.1 . . 3 𝑋 = βˆͺ 𝐽
2 cmphaushmeo.2 . . 3 π‘Œ = βˆͺ 𝐾
31, 2hmeof1o 23275 . 2 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
4 f1ocnv 6845 . . . . . . . 8 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘Œβ€“1-1-onto→𝑋)
5 f1of 6833 . . . . . . . 8 (◑𝐹:π‘Œβ€“1-1-onto→𝑋 β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
64, 5syl 17 . . . . . . 7 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
76a1i 11 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘ŒβŸΆπ‘‹))
8 f1orel 6836 . . . . . . . . . . . 12 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ Rel 𝐹)
98ad2antll 727 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ Rel 𝐹)
10 dfrel2 6188 . . . . . . . . . . 11 (Rel 𝐹 ↔ ◑◑𝐹 = 𝐹)
119, 10sylib 217 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ ◑◑𝐹 = 𝐹)
1211imaeq1d 6058 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (◑◑𝐹 β€œ π‘₯) = (𝐹 β€œ π‘₯))
13 simp2 1137 . . . . . . . . . . 11 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Haus)
1413adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ 𝐾 ∈ Haus)
15 imassrn 6070 . . . . . . . . . . 11 (𝐹 β€œ π‘₯) βŠ† ran 𝐹
16 f1ofo 6840 . . . . . . . . . . . . 13 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
1716ad2antll 727 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
18 forn 6808 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ ran 𝐹 = π‘Œ)
1917, 18syl 17 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ ran 𝐹 = π‘Œ)
2015, 19sseqtrid 4034 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (𝐹 β€œ π‘₯) βŠ† π‘Œ)
21 simpl3 1193 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
22 simp1 1136 . . . . . . . . . . . . 13 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Comp)
2322adantr 481 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ 𝐽 ∈ Comp)
24 simprl 769 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ π‘₯ ∈ (Clsdβ€˜π½))
25 cmpcld 22913 . . . . . . . . . . . 12 ((𝐽 ∈ Comp ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt π‘₯) ∈ Comp)
2623, 24, 25syl2anc 584 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (𝐽 β†Ύt π‘₯) ∈ Comp)
27 imacmp 22908 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 β†Ύt π‘₯) ∈ Comp) β†’ (𝐾 β†Ύt (𝐹 β€œ π‘₯)) ∈ Comp)
2821, 26, 27syl2anc 584 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (𝐾 β†Ύt (𝐹 β€œ π‘₯)) ∈ Comp)
292hauscmp 22918 . . . . . . . . . 10 ((𝐾 ∈ Haus ∧ (𝐹 β€œ π‘₯) βŠ† π‘Œ ∧ (𝐾 β†Ύt (𝐹 β€œ π‘₯)) ∈ Comp) β†’ (𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))
3014, 20, 28, 29syl3anc 1371 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))
3112, 30eqeltrd 2833 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (π‘₯ ∈ (Clsdβ€˜π½) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)) β†’ (◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))
3231expr 457 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ)))
3332ralrimdva 3154 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ)))
347, 33jcad 513 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (◑𝐹:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))))
35 haustop 22842 . . . . . . . 8 (𝐾 ∈ Haus β†’ 𝐾 ∈ Top)
3613, 35syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Top)
372toptopon 22426 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3836, 37sylib 217 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
39 cmptop 22906 . . . . . . . 8 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
4022, 39syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Top)
411toptopon 22426 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
4240, 41sylib 217 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
43 iscncl 22780 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (◑𝐹 ∈ (𝐾 Cn 𝐽) ↔ (◑𝐹:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))))
4438, 42, 43syl2anc 584 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (◑𝐹 ∈ (𝐾 Cn 𝐽) ↔ (◑𝐹:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))))
4534, 44sylibrd 258 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹 ∈ (𝐾 Cn 𝐽)))
46 simp3 1138 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
4745, 46jctild 526 . . 3 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◑𝐹 ∈ (𝐾 Cn 𝐽))))
48 ishmeo 23270 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◑𝐹 ∈ (𝐾 Cn 𝐽)))
4947, 48imbitrrdi 251 . 2 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹 ∈ (𝐽Homeo𝐾)))
503, 49impbid2 225 1 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-ontoβ†’π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  βˆͺ cuni 4908  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679  Rel wrel 5681  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   β†Ύt crest 17368  Topctop 22402  TopOnctopon 22419  Clsdccld 22527   Cn ccn 22735  Hauscha 22819  Compccmp 22897  Homeochmeo 23264
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-rest 17370  df-topgen 17391  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-cls 22532  df-cn 22738  df-haus 22826  df-cmp 22898  df-hmeo 23266
This theorem is referenced by:  cncfcnvcn  24448
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