Theorem cnmpt12f 23390

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))
cnmpt1t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
cnmpt12f.f	⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))

Assertion

Ref	Expression
cnmpt12f	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt12f

Step	Hyp	Ref	Expression
1		df-ov 7414	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	mpteq2i 5252	. 2 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩))
3		cnmptid.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		cnmpt11.a	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5		cnmpt1t.b	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
6	3, 4, 5	cnmpt1t 23389	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
7		cnmpt12f.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
8	3, 6, 7	cnmpt11f 23388	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩)) ∈ (𝐽 Cn 𝑀))
9	2, 8	eqeltrid 2835	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 2104  ⟨cop 4633   ↦ cmpt 5230  ‘cfv 6542  (class class class)co 7411  TopOnctopon 22632   Cn ccn 22948   ×_t ctx 23284

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cn 22951  df-tx 23286

This theorem is referenced by:  cnmpt12  23391  cnmpt1plusg  23811  istgp2  23815  clsnsg  23834  tgpt0  23843  cnmpt1vsca  23918  cnmpt1ds  24578  fsumcn  24608  expcn  24610  expcnOLD  24612  divccnOLD  24613  cncfmpt2f  24655  cdivcncf  24661  iirevcn  24671  iihalf1cnOLD  24674  iihalf2cn  24676  iihalf2cnOLD  24677  icchmeo  24685  icchmeoOLD  24686  evth  24705  evth2  24706  pcoass  24771  cnmpt1ip  24995  dvcnvlem  25728  plycnOLD  26011  psercn2  26171  atansopn  26673  efrlim  26710  ipasslem7  30356  occllem  30823  hmopidmchi  31671  cvxpconn  34531  cvmlift2lem2  34593  cvmlift2lem3  34594  cvmliftphtlem  34606  sinccvglem  34955  knoppcnlem10  35681  broucube  36825  areacirclem2  36880  fprodcnlem  44613