Theorem cmphaushmeo 23713

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:𝑋–1-1-onto→𝑌))

Proof of Theorem cmphaushmeo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmphaushmeo.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2		cmphaushmeo.2	. . 3 ⊢ 𝑌 = ∪ 𝐾
3	1, 2	hmeof1o 23677	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1ocnv 6775	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
5		f1of 6763	. . . . . . . 8 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌⟶𝑋)
7	6	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌⟶𝑋))
8		f1orel 6766	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Rel 𝐹)
9	8	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → Rel 𝐹)
10		dfrel2 6136	. . . . . . . . . . 11 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
11	9, 10	sylib 218	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → ◡◡𝐹 = 𝐹)
12	11	imaeq1d 6008	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥))
13		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
14	13	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐾 ∈ Haus)
15		imassrn 6020	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
16		f1ofo 6770	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
17	16	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹:𝑋–onto→𝑌)
18		forn 6738	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → ran 𝐹 = 𝑌)
20	15, 19	sseqtrid 3977	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐹 “ 𝑥) ⊆ 𝑌)
21		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐽 ∈ Comp)
24		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝑥 ∈ (Clsd‘𝐽))
25		cmpcld 23315	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑥) ∈ Comp)
26	23, 24, 25	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐽 ↾t 𝑥) ∈ Comp)
27		imacmp 23310	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑥) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp)
28	21, 26, 27	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp)
29	2	hauscmp 23320	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Haus ∧ (𝐹 “ 𝑥) ⊆ 𝑌 ∧ (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
30	14, 20, 28, 29	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
31	12, 30	eqeltrd 2831	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
32	31	expr 456	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐹:𝑋–1-1-onto→𝑌 → (◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾)))
33	32	ralrimdva 3132	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → ∀𝑥 ∈ (Clsd‘𝐽)(◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾)))
34	7, 33	jcad 512	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾))))
35		haustop 23244	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	13, 35	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	2	toptopon 22830	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
38	36, 37	sylib 218	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
39		cmptop 23308	. . . . . . . 8 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
40	22, 39	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
41	1	toptopon 22830	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
42	40, 41	sylib 218	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
43		iscncl 23182	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (◡𝐹 ∈ (𝐾 Cn 𝐽) ↔ (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾))))
44	38, 42, 43	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (◡𝐹 ∈ (𝐾 Cn 𝐽) ↔ (◡𝐹:𝑌⟶𝑋 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡◡𝐹 “ 𝑥) ∈ (Clsd‘𝐾))))
45	34, 44	sylibrd 259	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹 ∈ (𝐾 Cn 𝐽)))
46		simp3 1138	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
47	45, 46	jctild 525	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))))
48		ishmeo 23672	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))
49	47, 48	imbitrrdi 252	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ (𝐽Homeo𝐾)))
50	3, 49	impbid2 226	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-onto→𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3902  ∪ cuni 4859  ^◡ccnv 5615  ran crn 5617   “ cima 5619  Rel wrel 5621  ⟶wf 6477  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   ↾_t crest 17321  Topctop 22806  TopOnctopon 22823  Clsdccld 22929   Cn ccn 23137  Hauscha 23221  Compccmp 23299  Homeochmeo 23666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-1o 8385  df-2o 8386  df-map 8752  df-en 8870  df-dom 8871  df-fin 8873  df-fi 9295  df-rest 17323  df-topgen 17344  df-top 22807  df-topon 22824  df-bases 22859  df-cld 22932  df-cls 22934  df-cn 23140  df-haus 23228  df-cmp 23300  df-hmeo 23668

This theorem is referenced by:  cncfcnvcn  24844