Theorem cnmpt1t 22816

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))

Assertion

Ref	Expression
cnmpt1t	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt1t

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptid.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		toponuni 22063	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3		mpteq1 5167	. . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩))
5		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
6		cnmpt11.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7		cntop2 22392	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 22067	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		cnf2 22400	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
12	1, 10, 6, 11	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
13	12	fvmptelrn 6987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐾)
14		eqid 2738	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
15	14	fvmpt2 6886	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐾) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
16	5, 13, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
17		cnmpt1t.b	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
18		cntop2 22392	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
20		toptopon2 22067	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
21	19, 20	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
22		cnf2 22400	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
23	1, 21, 17, 22	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
24	23	fvmptelrn 6987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ∪ 𝐿)
25		eqid 2738	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
26	25	fvmpt2 6886	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
27	5, 24, 26	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
28	16, 27	opeq12d 4812	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
29	28	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
30	4, 29	eqtr3d 2780	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
31		eqid 2738	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
32		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩
33		nffvmpt1 6785	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦)
34		nffvmpt1 6785	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)
35	33, 34	nfop 4820	. . . . 5 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩
36		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦))
37		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦))
38	36, 37	opeq12d 4812	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩)
39	32, 35, 38	cbvmpt 5185	. . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑦 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩)
40	31, 39	txcnmpt 22775	. . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
41	6, 17, 40	syl2anc 584	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
42	30, 41	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567  ∪ cuni 4839   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-tx 22713

This theorem is referenced by:  cnmpt12f  22817  xkoinjcn  22838  txconn  22840  imasnopn  22841  imasncld  22842  imasncls  22843  ptunhmeo  22959  xkohmeo  22966  cnrehmeo  24116