Theorem cnmpt12f 23551

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 7352	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	mpteq2i 5188	. 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 23550	. . 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 23549	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩)) ∈ (𝐽 Cn 𝑀))
9	2, 8	eqeltrid 2832	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 2109  ⟨cop 4583   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  TopOnctopon 22795   Cn ccn 23109   ×_t ctx 23445

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-csb 3852  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-iun 4943  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-1st 7924  df-2nd 7925  df-map 8755  df-topgen 17347  df-top 22779  df-topon 22796  df-bases 22831  df-cn 23112  df-tx 23447

This theorem is referenced by:  cnmpt12  23552  cnmpt1plusg  23972  istgp2  23976  clsnsg  23995  tgpt0  24004  cnmpt1vsca  24079  cnmpt1ds  24729  fsumcn  24759  expcn  24761  expcnOLD  24763  divccnOLD  24764  cncfmpt2f  24806  cdivcncf  24812  iirevcn  24822  iihalf1cnOLD  24825  iihalf2cn  24827  iihalf2cnOLD  24828  icchmeo  24836  icchmeoOLD  24837  evth  24856  evth2  24857  pcoass  24922  cnmpt1ip  25145  dvcnvlem  25878  plycnOLD  26165  psercn2OLD  26331  atansopn  26840  efrlim  26877  efrlimOLD  26878  ipasslem7  30780  occllem  31247  hmopidmchi  32095  cvxpconn  35215  cvmlift2lem2  35277  cvmlift2lem3  35278  cvmliftphtlem  35290  sinccvglem  35645  broucube  37634  areacirclem2  37689