Theorem cnmpt1t 22273

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 21522	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3		mpteq1 5154	. . . 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 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
6		cnmpt11.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7		cntop2 21849	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 21526	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 220	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		cnf2 21857	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
12	1, 10, 6, 11	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
13	12	fvmptelrn 6877	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐾)
14		eqid 2821	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
15	14	fvmpt2 6779	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐾) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
16	5, 13, 15	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
17		cnmpt1t.b	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
18		cntop2 21849	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
20		toptopon2 21526	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
21	19, 20	sylib 220	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
22		cnf2 21857	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
23	1, 21, 17, 22	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
24	23	fvmptelrn 6877	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ∪ 𝐿)
25		eqid 2821	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
26	25	fvmpt2 6779	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
27	5, 24, 26	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
28	16, 27	opeq12d 4811	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
29	28	mpteq2dva 5161	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
30	4, 29	eqtr3d 2858	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
31		eqid 2821	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
32		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩
33		nffvmpt1 6681	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦)
34		nffvmpt1 6681	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)
35	33, 34	nfop 4819	. . . . 5 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩
36		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦))
37		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦))
38	36, 37	opeq12d 4811	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩)
39	32, 35, 38	cbvmpt 5167	. . . 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 22232	. . 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 586	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
42	30, 41	eqeltrrd 2914	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4573  ∪ cuni 4838   ↦ cmpt 5146  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Topctop 21501  TopOnctopon 21518   Cn ccn 21832   ×_t ctx 22168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-map 8408  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cn 21835  df-tx 22170

This theorem is referenced by:  cnmpt12f  22274  xkoinjcn  22295  txconn  22297  imasnopn  22298  imasncld  22299  imasncls  22300  ptunhmeo  22416  xkohmeo  22423  cnrehmeo  23557