Theorem rncmp 23371

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 2737	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2737	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23221	. . . . . 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 6663	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 Fn ∪ 𝐽)
7		dffn4 6752	. . . 4 ⊢ (𝐹 Fn ∪ 𝐽 ↔ 𝐹:∪ 𝐽–onto→ran 𝐹)
8	6, 7	sylib 218	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽–onto→ran 𝐹)
9		cntop2 23216	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	9	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
11	5	frnd 6670	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ∪ 𝐾)
12	3	restuni 23137	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ∪ 𝐾) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
13	10, 11, 12	syl2anc 585	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = ∪ (𝐾 ↾t ran 𝐹))
14		foeq3 6744	. . . 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 232	. 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 22893	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
19	10, 18	sylib 218	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
20		ssidd 3946	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 ⊆ ran 𝐹)
21		cnrest2 23261	. . . 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 1374	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹))))
23	17, 22	mpbid 232	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)))
24		eqid 2737	. . 3 ⊢ ∪ (𝐾 ↾t ran 𝐹) = ∪ (𝐾 ↾t ran 𝐹)
25	24	cncmp 23367	. 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 1374	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ∪ cuni 4851  ran crn 5625   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  Topctop 22868  TopOnctopon 22885   Cn ccn 23199  Compccmp 23361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8398  df-map 8768  df-en 8887  df-dom 8888  df-fin 8890  df-fi 9317  df-rest 17376  df-topgen 17397  df-top 22869  df-topon 22886  df-bases 22921  df-cn 23202  df-cmp 23362

This theorem is referenced by:  imacmp  23372  kgencn2  23532  bndth  24935