Theorem cnmpt12f 21218

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 6527	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	mpteq2i 4660	. 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 21217	. . 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 21216	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩)) ∈ (𝐽 Cn 𝑀))
9	2, 8	syl5eqel 2688	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 1976  ⟨cop 4127   ↦ cmpt 4634  ‘cfv 5787  (class class class)co 6524  TopOnctopon 20457   Cn ccn 20777   ×_t ctx 21112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-map 7720  df-topgen 15870  df-top 20460  df-bases 20461  df-topon 20462  df-cn 20780  df-tx 21114

This theorem is referenced by:  cnmpt12  21219  cnmpt1plusg  21640  istgp2  21644  clsnsg  21662  tgpt0  21671  cnmpt1vsca  21746  cnmpt1ds  22382  fsumcn  22409  expcn  22411  divccn  22412  cncfmpt2f  22453  cdivcncf  22456  iirevcn  22465  iihalf1cn  22467  iihalf2cn  22469  icchmeo  22476  evth  22494  evth2  22495  pcoass  22560  cnmpt1ip  22769  dvcnvlem  23457  plycn  23735  psercn2  23895  atansopn  24373  efrlim  24410  ipasslem7  26878  occllem  27349  hmopidmchi  28197  cvxpcon  30281  cvmlift2lem2  30343  cvmlift2lem3  30344  cvmliftphtlem  30356  sinccvglem  30623  knoppcnlem10  31465  broucube  32413  areacirclem2  32471  fprodcnlem  38467