Theorem cnmpt21f 22274

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 21842	. . . 4 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		toptopon2 21520	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
8	6, 7	sylib 220	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
9		eqid 2821	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
10		eqid 2821	. . . . . 6 ⊢ ∪ 𝑀 = ∪ 𝑀
11	9, 10	cnf 21848	. . . . 5 ⊢ (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹:∪ 𝐿⟶∪ 𝑀)
12	4, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:∪ 𝐿⟶∪ 𝑀)
13	12	feqmptd 6727	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)))
14	13, 4	eqeltrrd 2914	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿 ↦ (𝐹‘𝑧)) ∈ (𝐿 Cn 𝑀))
15		fveq2 6664	. 2 ⊢ (𝑧 = 𝐴 → (𝐹‘𝑧) = (𝐹‘𝐴))
16	1, 2, 3, 8, 14, 15	cnmpt21 22273	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ∈ wcel 2110  ∪ cuni 4831   ↦ cmpt 5138  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Topctop 21495  TopOnctopon 21512   Cn ccn 21826   ×_t ctx 22162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-tx 22164

This theorem is referenced by:  cnmpt22  22276  cnmptk2  22288  txhmeo  22405  tgpsubcn  22692  istgp2  22693  dvrcn  22786  htpyid  23575  htpyco1  23576  reparphti  23595  pcocn  23615  pcorevlem  23624  cxpcn  25320  dipcn  28491  mndpluscn  31164  cvxsconn  32485  cvmlift2lem6  32550  cvmlift2lem12  32556