Theorem connima 23454

Description: The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
connima.x	⊢ 𝑋 = ∪ 𝐽
connima.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
connima.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
connima.c	⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn)

Assertion

Ref	Expression
connima	⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn)

Proof of Theorem connima

Step	Hyp	Ref	Expression
1		connima.c	. 2 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn)
2		connima.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		connima.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4		eqid 2740	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23275	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
7	6	ffund 6751	. . . 4 ⊢ (𝜑 → Fun 𝐹)
8		connima.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
9	6	fdmd 6757	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋)
10	8, 9	sseqtrrd 4050	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
11		fores 6844	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
12	7, 10, 11	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
13		cntop2 23270	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
14	2, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Top)
15		imassrn 6100	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
16	6	frnd 6755	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐾)
17	15, 16	sstrid 4020	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
18	4	restuni 23191	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)))
19	14, 17, 18	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)))
20		foeq3 6832	. . . 4 ⊢ ((𝐹 “ 𝐴) = ∪ (𝐾 ↾t (𝐹 “ 𝐴)) → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))))
21	19, 20	syl 17	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴))))
22	12, 21	mpbid 232	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴)))
23	3	cnrest 23314	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
24	2, 8, 23	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
25		toptopon2 22945	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
26	14, 25	sylib 218	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
27		df-ima 5713	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
28		eqimss2 4068	. . . . 5 ⊢ ((𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) → ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴))
29	27, 28	mp1i 13	. . . 4 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴))
30		cnrest2 23315	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran (𝐹 ↾ 𝐴) ⊆ (𝐹 “ 𝐴) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))))
31	26, 29, 17, 30	syl3anc 1371	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))))
32	24, 31	mpbid 232	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴))))
33		eqid 2740	. . 3 ⊢ ∪ (𝐾 ↾t (𝐹 “ 𝐴)) = ∪ (𝐾 ↾t (𝐹 “ 𝐴))
34	33	cnconn 23451	. 2 ⊢ (((𝐽 ↾t 𝐴) ∈ Conn ∧ (𝐹 ↾ 𝐴):𝐴–onto→∪ (𝐾 ↾t (𝐹 “ 𝐴)) ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t (𝐹 “ 𝐴)))) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn)
35	1, 22, 32, 34	syl3anc 1371	1 ⊢ (𝜑 → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Conn)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∪ cuni 4931  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  TopOnctopon 22937   Cn ccn 23253  Conncconn 23440

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-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-map 8886  df-en 9004  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-cn 23256  df-conn 23441

This theorem is referenced by:  tgpconncompeqg  24141  tgpconncomp  24142