Theorem cnmpt1t 23016

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 22263	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3		mpteq1 5198	. . . 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 22592	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 22267	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		cnf2 22600	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
12	1, 10, 6, 11	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
13	12	fvmptelcdm 7061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐾)
14		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
15	14	fvmpt2 6959	. . . . . 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 22592	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
20		toptopon2 22267	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
21	19, 20	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
22		cnf2 22600	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
23	1, 21, 17, 22	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
24	23	fvmptelcdm 7061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ∪ 𝐿)
25		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
26	25	fvmpt2 6959	. . . . . 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 4838	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
29	28	mpteq2dva 5205	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
30	4, 29	eqtr3d 2778	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
31		eqid 2736	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
32		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩
33		nffvmpt1 6853	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦)
34		nffvmpt1 6853	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)
35	33, 34	nfop 4846	. . . . 5 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩
36		fveq2 6842	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦))
37		fveq2 6842	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦))
38	36, 37	opeq12d 4838	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩)
39	32, 35, 38	cbvmpt 5216	. . . 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 22975	. . 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 2839	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4592  ∪ cuni 4865   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Topctop 22242  TopOnctopon 22259   Cn ccn 22575   ×_t ctx 22911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-tx 22913

This theorem is referenced by:  cnmpt12f  23017  xkoinjcn  23038  txconn  23040  imasnopn  23041  imasncld  23042  imasncls  23043  ptunhmeo  23159  xkohmeo  23166  cnrehmeo  24316