Theorem rncmp 22455

Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
rncmp	⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp)

Proof of Theorem rncmp

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Comp)
2		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 22305	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	4	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffnd 6585	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
7		dffn4 6678	. . . 4 ⊢ (𝐹 Fn ∪ 𝐽 ↔ 𝐹:∪ 𝐽–onto→ran 𝐹)
8	6, 7	sylib 217	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→ran 𝐹)
9		cntop2 22300	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	9	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
11	5	frnd 6592	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ∪ 𝐾)
12	3	restuni 22221	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
13	10, 11, 12	syl2anc 583	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
14		foeq3 6670	. . . 4 ⊢ (ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹) → (𝐹:∪ 𝐽–onto→ran 𝐹 ↔ 𝐹:∪ 𝐽–onto→∪ (𝐾 ↾t ran 𝐹)))
15	13, 14	syl 17	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:∪ 𝐽–onto→ran 𝐹 ↔ 𝐹:∪ 𝐽–onto→∪ (𝐾 ↾t ran 𝐹)))
16	8, 15	mpbid 231	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→∪ (𝐾 ↾t ran 𝐹))
17		simpr 484	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn 𝐾))
18		toptopon2 21975	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	10, 18	sylib 217	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3940	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 22345	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ ∪ 𝐾) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
22	19, 20, 11, 21	syl3anc 1369	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
23	17, 22	mpbid 231	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
24		eqid 2738	. . 3 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cncmp 22451	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹:∪ 𝐽–onto→∪ (𝐾 ↾t ran 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))) → (𝐾 ↾t ran 𝐹) ∈ Comp)
26	1, 16, 23, 25	syl3anc 1369	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ∪ cuni 4836  ran crn 5581   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  TopOnctopon 21967   Cn ccn 22283  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cn 22286  df-cmp 22446

This theorem is referenced by:  imacmp  22456  kgencn2  22616  bndth  24027