Theorem imacmp 23445

Description: The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
imacmp	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Proof of Theorem imacmp

Step	Hyp	Ref	Expression
1		df-ima 5656	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2	1	oveq2i 7402	. 2 ⊢ (𝐾 ↾t (𝐹 “ 𝐴)) = (𝐾 ↾t ran (𝐹 ↾ 𝐴))
3		simpr 488	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Comp)
4		simpl 486	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		inss2 4187	. . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		eqid 2761	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cnrest 23333	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
8	4, 5, 7	sylancl 595	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
9		resdmres 6214	. . . . 5 ⊢ (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ 𝐴)
10		dmres 5994	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
11		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
12	6, 11	cnf 23294	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
13		fdm 6696	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
14	4, 12, 13	3syl 18	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom 𝐹 = ∪ 𝐽)
15	14	ineq2d 4170	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐴 ∩ dom 𝐹) = (𝐴 ∩ ∪ 𝐽))
16	10, 15	eqtrid 2808	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → dom (𝐹 ↾ 𝐴) = (𝐴 ∩ ∪ 𝐽))
17	16	reseq2d 5961	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ dom (𝐹 ↾ 𝐴)) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
18	9, 17	eqtr3id 2810	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) = (𝐹 ↾ (𝐴 ∩ ∪ 𝐽)))
19		cmptop 23443	. . . . . . 7 ⊢ ((𝐽 ↾t 𝐴) ∈ Comp → (𝐽 ↾t 𝐴) ∈ Top)
20	19	adantl 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) ∈ Top)
21		restrcl 23205	. . . . . 6 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
22	6	restin 23214	. . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
23	20, 21, 22	3syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)))
24	23	oveq1d 7406	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → ((𝐽 ↾t 𝐴) Cn 𝐾) = ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) Cn 𝐾))
25	8, 18, 24	3eltr4d 2876	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
26		rncmp 23444	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ Comp ∧ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
27	3, 25, 26	syl2anc 593	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t ran (𝐹 ↾ 𝐴)) ∈ Comp)
28	2, 27	eqeltrid 2865	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝐴) ∈ Comp) → (𝐾 ↾t (𝐹 “ 𝐴)) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4862  dom cdm 5643  ran crn 5644   ↾ cres 5645   “ cima 5646  ⟶wf 6512  (class class class)co 7391   ↾_t crest 17440  Topctop 22941   Cn ccn 23272  Compccmp 23434

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 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

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 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-1o 8431  df-map 8804  df-en 8922  df-dom 8923  df-fin 8925  df-fi 9351  df-rest 17442  df-topgen 17463  df-top 22942  df-topon 22959  df-bases 22994  df-cn 23275  df-cmp 23435

This theorem is referenced by:  kgencn3  23606  txkgen  23700  xkoco1cn  23705  xkococnlem  23707  cmphaushmeo  23848  cnheiborlem  25004