Theorem cnmpt11f 23629

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 23206	. . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝐾 ∈ Top)
5		toptopon2 22883	. . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
7		cnmpt11f.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))
8		eqid 2736	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9		eqid 2736	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
10	8, 9	cnf 23211	. . . . 5 ⊢ (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹:∪ 𝐾⟶∪ 𝐿)
11	7, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐾⟶∪ 𝐿)
12	11	feqmptd 6908	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)))
13	12, 7	eqeltrrd 2837	. 2 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝐹‘𝑦)) ∈ (𝐾 Cn 𝐿))
14		fveq2 6840	. 2 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
15	1, 2, 6, 13, 14	cnmpt11 23628	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∈ wcel 2114  ∪ cuni 4850   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Topctop 22858  TopOnctopon 22875   Cn ccn 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-top 22859  df-topon 22876  df-cn 23192

This theorem is referenced by:  cnmpt12f  23631  tgpmulg  24058  prdstgpd  24090  pcorevcl  24992  pcorevlem  24993  logcn  26611  loglesqrt  26725  efrlim  26933  cvmliftlem8  35474  knoppcnlem10  36762  areacirclem2  38030  areacirclem4  38032