Theorem cnmpt22f 21526

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))
cnmpt2t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
cnmpt22f.f	⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))

Assertion

Ref	Expression
cnmpt22f	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

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

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

Proof of Theorem cnmpt22f

Dummy variables 𝑤 𝑧 are mutually distinct and 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		cnmpt2t.b	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
5		cntop2 21093	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		eqid 2651	. . . 4 ⊢ ∪ 𝐿 = ∪ 𝐿
8	7	toptopon 20770	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
9	6, 8	sylib 208	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
10		cntop2 21093	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
11	4, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑀 ∈ Top)
12		eqid 2651	. . . 4 ⊢ ∪ 𝑀 = ∪ 𝑀
13	12	toptopon 20770	. . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
14	11, 13	sylib 208	. 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
15		txtopon 21442	. . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
16	9, 14, 15	syl2anc 694	. . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
17		cnmpt22f.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
18		cntop2 21093	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top)
20		eqid 2651	. . . . . . . 8 ⊢ ∪ 𝑁 = ∪ 𝑁
21	20	toptopon 20770	. . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
22	19, 21	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
23		cnf2 21101	. . . . . 6 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
24	16, 22, 17, 23	syl3anc 1366	. . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
25		ffn 6083	. . . . 5 ⊢ (𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀))
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀))
27		fnov 6810	. . . 4 ⊢ (𝐹 Fn (∪ 𝐿 × ∪ 𝑀) ↔ 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
28	26, 27	sylib 208	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
29	28, 17	eqeltrrd 2731	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30		oveq12 6699	. 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
31	1, 2, 3, 4, 9, 14, 29, 30	cnmpt22 21525	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ∪ cuni 4468   × cxp 5141   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  Topctop 20746  TopOnctopon 20763   Cn ccn 21076   ×_t ctx 21411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-tx 21413

This theorem is referenced by:  cnmptcom  21529  cnmpt2plusg  21939  istgp2  21942  cnmpt2vsca  22045  cnmpt2ds  22693  divcn  22718  cnrehmeo  22799  htpycom  22822  htpyco1  22824  htpycc  22826  reparphti  22843  pcohtpylem  22865  cnmpt2ip  23093  cxpcn  24531  vmcn  27682  dipcn  27703  mndpluscn  30100  cvxsconn  31351