Theorem cnmpt21f 21846

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
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt21f.f	⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀))

Assertion

Ref	Expression
cnmpt21f	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt21f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmpt21.j	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmpt21.k	. 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		cnmpt21.a	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4		cnmpt21f.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀))
5		cntop1 21415	. . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		eqid 2825	. . . 4 ⊢ ∪ 𝐿 = ∪ 𝐿
8	7	toptopon 21092	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
9	6, 8	sylib 210	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
10		eqid 2825	. . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀
11	7, 10	cnf 21421	. . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀)
12	4, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀)
13	12	feqmptd 6496	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)))
14	13, 4	eqeltrrd 2907	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀))
15		fveq2 6433	. 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
16	1, 2, 3, 9, 14, 15	cnmpt21 21845	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 2166  ∪ cuni 4658   ↦ cmpt 4952  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907  Topctop 21068  TopOnctopon 21085   Cn ccn 21399   ×_t ctx 21734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-topgen 16457  df-top 21069  df-topon 21086  df-bases 21121  df-cn 21402  df-tx 21736

This theorem is referenced by:  cnmpt22  21848  cnmptk2  21860  txhmeo  21977  tgpsubcn  22264  istgp2  22265  dvrcn  22357  htpyid  23146  htpyco1  23147  reparphti  23166  pcocn  23186  pcorevlem  23195  cxpcn  24888  dipcn  28130  mndpluscn  30517  cvxsconn  31771  cvmlift2lem6  31836  cvmlift2lem12  31842