Theorem cnco 23210

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 23184	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cntop2 23185	. . 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 2736	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
5		eqid 2736	. . . . 5 ⊢ ∪ 𝐿 = ∪ 𝐿
6	4, 5	cnf 23190	. . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐺:∪ 𝐾⟶∪ 𝐿)
7		eqid 2736	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7, 4	cnf 23190	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fco 6686	. . . 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 5834	. . . . . . 7 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
12	11	imaeq1i 6016	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
13		imaco 6209	. . . . . 6 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
14	12, 13	eqtri 2759	. . . . 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 23209	. . . . . . 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 23209	. . . . . 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 2840	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)
21	20	ralrimiva 3128	. . 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 23182	. 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 2113  ∀wral 3051  ∪ cuni 4863  ^◡ccnv 5623   “ cima 5627   ∘ ccom 5628  ⟶wf 6488  (class class class)co 7358  Topctop 22837   Cn ccn 23168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  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-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-top 22838  df-topon 22855  df-cn 23171

This theorem is referenced by:  kgencn2  23501  txcn  23570  xkoco1cn  23601  xkoco2cn  23602  xkococnlem  23603  xkococn  23604  cnmpt11  23607  cnmpt21  23615  hmeoco  23716  qtophmeo  23761  htpyco1  24933  htpyco2  24934  phtpyco2  24945  reparphti  24952  reparphtiOLD  24953  reparpht  24954  phtpcco2  24955  copco  24974  pi1cof  25015  pi1coghm  25017  cnpconn  35424  txsconnlem  35434  txsconn  35435  cvmlift3lem2  35514  cvmlift3lem4  35516  cvmlift3lem5  35517  cvmlift3lem6  35518  hausgraph  43443