Theorem cmphaushmeo 23840

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 23804	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1ocnv 6815	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
5		f1of 6802	. . . . . . . 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 6805	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Rel 𝐹)
9	8	ad2antll 739	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → Rel 𝐹)
10		dfrel2 6171	. . . . . . . . . . 11 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
11	9, 10	sylib 220	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → ◡◡𝐹 = 𝐹)
12	11	imaeq1d 6045	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥))
13		simp2 1149	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
14	13	adantr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐾 ∈ Haus)
15		imassrn 6057	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
16		f1ofo 6810	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
17	16	ad2antll 739	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹:𝑋–onto→𝑌)
18		forn 6777	. . . . . . . . . . . 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 3978	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐹 “ 𝑥) ⊆ 𝑌)
21		simpl3 1206	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22		simp1 1148	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
23	22	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐽 ∈ Comp)
24		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝑥 ∈ (Clsd‘𝐽))
25		cmpcld 23442	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑥) ∈ Comp)
26	23, 24, 25	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐽 ↾t 𝑥) ∈ Comp)
27		imacmp 23437	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑥) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp)
28	21, 26, 27	syl2anc 593	. . . . . . . . . 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 23447	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Haus ∧ (𝐹 “ 𝑥) ⊆ 𝑌 ∧ (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
30	14, 20, 28, 29	syl3anc 1389	. . . . . . . . 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 2861	. . . . . . . 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 460	. . . . . . 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 3161	. . . . . 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 520	. . . . 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 23371	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	13, 35	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	2	toptopon 22957	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
38	36, 37	sylib 220	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
39		cmptop 23435	. . . . . . . 8 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
40	22, 39	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
41	1	toptopon 22957	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
42	40, 41	sylib 220	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
43		iscncl 23309	. . . . . 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 593	. . . . 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 261	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹 ∈ (𝐾 Cn 𝐽)))
46		simp3 1150	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
47	45, 46	jctild 533	. . 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 23799	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))
49	47, 48	imbitrrdi 254	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ (𝐽Homeo𝐾)))
50	3, 49	impbid2 228	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-onto→𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3904  ∪ cuni 4864  ^◡ccnv 5644  ran crn 5646   “ cima 5648  Rel wrel 5650  ⟶wf 6513  –onto→wfo 6515  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392   ↾_t crest 17432  Topctop 22933  TopOnctopon 22950  Clsdccld 23056   Cn ccn 23264  Hauscha 23348  Compccmp 23426  Homeochmeo 23793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-1o 8432  df-2o 8433  df-map 8805  df-en 8924  df-dom 8925  df-fin 8927  df-fi 9354  df-rest 17434  df-topgen 17455  df-top 22934  df-topon 22951  df-bases 22986  df-cld 23059  df-cls 23061  df-cn 23267  df-haus 23355  df-cmp 23427  df-hmeo 23795

This theorem is referenced by:  cncfcnvcn  24967