Theorem cmphaushmeo 23829

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 23793	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1ocnv 6874	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
5		f1of 6862	. . . . . . . 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 6865	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Rel 𝐹)
9	8	ad2antll 728	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → Rel 𝐹)
10		dfrel2 6220	. . . . . . . . . . 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 6088	. . . . . . . . 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 6100	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
16		f1ofo 6869	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
17	16	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹:𝑋–onto→𝑌)
18		forn 6837	. . . . . . . . . . . 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 4061	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐹 “ 𝑥) ⊆ 𝑌)
21		simpl3 1193	. . . . . . . . . . 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 23431	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑥) ∈ Comp)
26	23, 24, 25	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐽 ↾t 𝑥) ∈ Comp)
27		imacmp 23426	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑥) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp)
28	21, 26, 27	syl2anc 583	. . . . . . . . . 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 23436	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Haus ∧ (𝐹 “ 𝑥) ⊆ 𝑌 ∧ (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
30	14, 20, 28, 29	syl3anc 1371	. . . . . . . . 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 2844	. . . . . . . 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 3160	. . . . . 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 23360	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	13, 35	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	2	toptopon 22944	. . . . . . 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 23424	. . . . . . . 8 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
40	22, 39	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
41	1	toptopon 22944	. . . . . . 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 23298	. . . . . 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 583	. . . . 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 23788	. . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ∪ cuni 4931  ^◡ccnv 5699  ran crn 5701   “ cima 5703  Rel wrel 5705  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  TopOnctopon 22937  Clsdccld 23045   Cn ccn 23253  Hauscha 23337  Compccmp 23415  Homeochmeo 23782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-map 8886  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480  df-rest 17482  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-cls 23050  df-cn 23256  df-haus 23344  df-cmp 23416  df-hmeo 23784

This theorem is referenced by:  cncfcnvcn  24971