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Theorem cmphaushmeo 23304
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 23268 . 2 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
4 f1ocnv 6846 . . . . . . . 8 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘Œβ€“1-1-onto→𝑋)
5 f1of 6834 . . . . . . . 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 6837 . . . . . . . . . . . 12 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ Rel 𝐹)
98ad2antll 728 . . . . . . . . . . 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 6189 . . . . . . . . . . 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 6059 . . . . . . . . 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 1138 . . . . . . . . . . 11 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Haus)
1413adantr 482 . . . . . . . . . 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 6071 . . . . . . . . . . 11 (𝐹 β€œ π‘₯) βŠ† ran 𝐹
16 f1ofo 6841 . . . . . . . . . . . . 13 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
1716ad2antll 728 . . . . . . . . . . . 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 6809 . . . . . . . . . . . 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 4035 . . . . . . . . . 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 1194 . . . . . . . . . . 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 1137 . . . . . . . . . . . . 13 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Comp)
2322adantr 482 . . . . . . . . . . . 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 770 . . . . . . . . . . . 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 22906 . . . . . . . . . . . 12 ((𝐽 ∈ Comp ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (𝐽 β†Ύt π‘₯) ∈ Comp)
2623, 24, 25syl2anc 585 . . . . . . . . . . 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 22901 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 β†Ύt π‘₯) ∈ Comp) β†’ (𝐾 β†Ύt (𝐹 β€œ π‘₯)) ∈ Comp)
2821, 26, 27syl2anc 585 . . . . . . . . . 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 22911 . . . . . . . . . 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 1372 . . . . . . . . 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 2834 . . . . . . . 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 458 . . . . . . 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 3155 . . . . . 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 514 . . . . 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 22835 . . . . . . . 8 (𝐾 ∈ Haus β†’ 𝐾 ∈ Top)
3613, 35syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐾 ∈ Top)
372toptopon 22419 . . . . . . 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 22899 . . . . . . . 8 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
4022, 39syl 17 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Top)
411toptopon 22419 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
4240, 41sylib 217 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
43 iscncl 22773 . . . . . 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 585 . . . . 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 259 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹 ∈ (𝐾 Cn 𝐽)))
46 simp3 1139 . . . 4 ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
4745, 46jctild 527 . . 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 23263 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◑𝐹 ∈ (𝐾 Cn 𝐽)))
4947, 48syl6ibr 252 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  βˆͺ cuni 4909  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680  Rel wrel 5682  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  TopOnctopon 22412  Clsdccld 22520   Cn ccn 22728  Hauscha 22812  Compccmp 22890  Homeochmeo 23257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-cls 22525  df-cn 22731  df-haus 22819  df-cmp 22891  df-hmeo 23259
This theorem is referenced by:  cncfcnvcn  24441
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