Theorem cnco 23160

Description: The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnco	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))

Proof of Theorem cnco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop1 23134	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cntop2 23135	. . 3 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
3	1, 2	anim12i 613	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top))
4		eqid 2730	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
5		eqid 2730	. . . . 5 ⊢ ∪ 𝐿 = ∪ 𝐿
6	4, 5	cnf 23140	. . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐺:∪ 𝐾⟶∪ 𝐿)
7		eqid 2730	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7, 4	cnf 23140	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fco 6715	. . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
10	6, 8, 9	syl2anr 597	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
11		cnvco 5852	. . . . . . 7 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
12	11	imaeq1i 6031	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
13		imaco 6227	. . . . . 6 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
14	12, 13	eqtri 2753	. . . . 5 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
15		simpll 766	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐹 ∈ (𝐽 Cn 𝐾))
16		cnima 23159	. . . . . . 7 ⊢ ((𝐺 ∈ (𝐾 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾)
17	16	adantll 714	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐺 “ 𝑥) ∈ 𝐾)
18		cnima 23159	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (◡𝐺 “ 𝑥) ∈ 𝐾) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)
19	15, 17, 18	syl2anc 584	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝐹 “ (◡𝐺 “ 𝑥)) ∈ 𝐽)
20	14, 19	eqeltrid 2833	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)
21	20	ralrimiva 3126	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)
22	10, 21	jca 511	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽))
23	7, 5	iscn2 23132	. 2 ⊢ ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿) ↔ ((𝐽 ∈ Top ∧ 𝐿 ∈ Top) ∧ ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)))
24	3, 22, 23	sylanbrc 583	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  ∪ cuni 4874  ^◡ccnv 5640   “ cima 5644   ∘ ccom 5645  ⟶wf 6510  (class class class)co 7390  Topctop 22787   Cn ccn 23118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-top 22788  df-topon 22805  df-cn 23121

This theorem is referenced by:  kgencn2  23451  txcn  23520  xkoco1cn  23551  xkoco2cn  23552  xkococnlem  23553  xkococn  23554  cnmpt11  23557  cnmpt21  23565  hmeoco  23666  qtophmeo  23711  htpyco1  24884  htpyco2  24885  phtpyco2  24896  reparphti  24903  reparphtiOLD  24904  reparpht  24905  phtpcco2  24906  copco  24925  pi1cof  24966  pi1coghm  24968  cnpconn  35224  txsconnlem  35234  txsconn  35235  cvmlift3lem2  35314  cvmlift3lem4  35316  cvmlift3lem5  35317  cvmlift3lem6  35318  hausgraph  43201