Theorem cnmpt21f 23566

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 23134	. . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		toptopon2 22812	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
8	6, 7	sylib 218	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
9		eqid 2730	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
10		eqid 2730	. . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀
11	9, 10	cnf 23140	. . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀)
12	4, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀)
13	12	feqmptd 6932	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)))
14	13, 4	eqeltrrd 2830	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀))
15		fveq2 6861	. 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
16	1, 2, 3, 8, 14, 15	cnmpt21 23565	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 2109  ∪ cuni 4874   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  Topctop 22787  TopOnctopon 22804   Cn ccn 23118   ×_t ctx 23454

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cn 23121  df-tx 23456

This theorem is referenced by:  cnmpt22  23568  cnmptk2  23580  txhmeo  23697  tgpsubcn  23984  istgp2  23985  dvrcn  24078  htpyid  24883  htpyco1  24884  reparphti  24903  reparphtiOLD  24904  pcocn  24924  pcorevlem  24933  cxpcn  26661  cxpcnOLD  26662  dipcn  30656  mndpluscn  33923  cvxsconn  35237  cvmlift2lem6  35302  cvmlift2lem12  35308