Theorem cmphaushmeo 24099

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 24063	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1ocnv 6829	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
5		f1of 6816	. . . . . . . 8 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
6	4, 5	syl 18	. . . . . . 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 6819	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Rel 𝐹)
9	8	ad2antll 742	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → Rel 𝐹)
10		dfrel2 6180	. . . . . . . . . . 11 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
11	9, 10	sylib 221	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → ◡◡𝐹 = 𝐹)
12	11	imaeq1d 6053	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥))
13		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
14	13	adantr 486	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐾 ∈ Haus)
15		imassrn 6065	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
16		f1ofo 6824	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
17	16	ad2antll 742	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹:𝑋–onto→𝑌)
18		forn 6791	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
19	17, 18	syl 18	. . . . . . . . . . 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 3973	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐹 “ 𝑥) ⊆ 𝑌)
21		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐹 ∈ (𝐽 Cn 𝐾))
22		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
23	22	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝐽 ∈ Comp)
24		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → 𝑥 ∈ (Clsd‘𝐽))
25		cmpcld 23700	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝑥) ∈ Comp)
26	23, 24, 25	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝐹:𝑋–1-1-onto→𝑌)) → (𝐽 ↾t 𝑥) ∈ Comp)
27		imacmp 23695	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑥) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp)
28	21, 26, 27	syl2anc 596	. . . . . . . . . 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 23705	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Haus ∧ (𝐹 “ 𝑥) ⊆ 𝑌 ∧ (𝐾 ↾t (𝐹 “ 𝑥)) ∈ Comp) → (𝐹 “ 𝑥) ∈ (Clsd‘𝐾))
30	14, 20, 28, 29	syl3anc 1398	. . . . . . . . 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 462	. . . . . . 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 3163	. . . . . 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 522	. . . . 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 23629	. . . . . . . 8 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	13, 35	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	2	toptopon 23215	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
38	36, 37	sylib 221	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
39		cmptop 23693	. . . . . . . 8 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
40	22, 39	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
41	1	toptopon 23215	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
42	40, 41	sylib 221	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
43		iscncl 23567	. . . . . 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 596	. . . . 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 262	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹 ∈ (𝐾 Cn 𝐽)))
46		simp3 1156	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
47	45, 46	jctild 535	. . 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 24058	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)))
49	47, 48	imbitrrdi 255	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ (𝐽Homeo𝐾)))
50	3, 49	impbid2 229	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-onto→𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5650  ran crn 5652   “ cima 5654  Rel wrel 5656  ⟶wf 6527  –onto→wfo 6529  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  TopOnctopon 23208  Clsdccld 23314   Cn ccn 23522  Hauscha 23606  Compccmp 23684  Homeochmeo 24052

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-1o 8460  df-2o 8461  df-map 8833  df-en 8958  df-dom 8959  df-fin 8961  df-fi 9387  df-rest 17573  df-topgen 17594  df-top 23192  df-topon 23209  df-bases 23244  df-cld 23317  df-cls 23319  df-cn 23525  df-haus 23613  df-cmp 23685  df-hmeo 24054

This theorem is used by:  cncfcnvcn  25226