Theorem cnco 23151

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 23125	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cntop2 23126	. . 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 2729	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
5		eqid 2729	. . . . 5 ⊢ ∪ 𝐿 = ∪ 𝐿
6	4, 5	cnf 23131	. . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐿) → 𝐺:∪ 𝐾⟶∪ 𝐿)
7		eqid 2729	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7, 4	cnf 23131	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fco 6676	. . . 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 5828	. . . . . . 7 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺)
12	11	imaeq1i 6008	. . . . . 6 ⊢ (◡(𝐺 ∘ 𝐹) “ 𝑥) = ((◡𝐹 ∘ ◡𝐺) “ 𝑥)
13		imaco 6200	. . . . . 6 ⊢ ((◡𝐹 ∘ ◡𝐺) “ 𝑥) = (◡𝐹 “ (◡𝐺 “ 𝑥))
14	12, 13	eqtri 2752	. . . . 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 23150	. . . . . . 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 23150	. . . . . 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 2832	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡(𝐺 ∘ 𝐹) “ 𝑥) ∈ 𝐽)
21	20	ralrimiva 3121	. . 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 23123	. 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 3044  ∪ cuni 4858  ^◡ccnv 5618   “ cima 5622   ∘ ccom 5623  ⟶wf 6478  (class class class)co 7349  Topctop 22778   Cn ccn 23109

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-top 22779  df-topon 22796  df-cn 23112

This theorem is referenced by:  kgencn2  23442  txcn  23511  xkoco1cn  23542  xkoco2cn  23543  xkococnlem  23544  xkococn  23545  cnmpt11  23548  cnmpt21  23556  hmeoco  23657  qtophmeo  23702  htpyco1  24875  htpyco2  24876  phtpyco2  24887  reparphti  24894  reparphtiOLD  24895  reparpht  24896  phtpcco2  24897  copco  24916  pi1cof  24957  pi1coghm  24959  cnpconn  35223  txsconnlem  35233  txsconn  35234  cvmlift3lem2  35313  cvmlift3lem4  35315  cvmlift3lem5  35316  cvmlift3lem6  35317  hausgraph  43198