Theorem cnmpt11f 23518

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptid.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt11f.f	⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))

Assertion

Ref	Expression
cnmpt11f	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿))

Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem cnmpt11f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptid.j	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmpt11.a	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
3		cntop2 23095	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐾 ∈ Top)
5		toptopon2 22770	. . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
6	4, 5	sylib 217	. 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
7		cnmpt11f.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))
8		eqid 2726	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9		eqid 2726	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
10	8, 9	cnf 23100	. . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿)
11	7, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿)
12	11	feqmptd 6953	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)))
13	12, 7	eqeltrrd 2828	. 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿))
14		fveq2 6884	. 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
15	1, 2, 6, 13, 14	cnmpt11 23517	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∈ wcel 2098  ∪ cuni 4902   ↦ cmpt 5224  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  Topctop 22745  TopOnctopon 22762   Cn ccn 23078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-top 22746  df-topon 22763  df-cn 23081

This theorem is referenced by:  cnmpt12f  23520  tgpmulg  23947  prdstgpd  23979  pcorevcl  24902  pcorevlem  24903  logcn  26531  loglesqrt  26643  efrlim  26851  efrlimOLD  26852  cvmliftlem8  34810  knoppcnlem10  35885  areacirclem2  37089  areacirclem4  37091