Theorem cnmpt1t 14872

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 14602	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3		mpteq1 4144	. . . 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 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩))
5		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
6		cnmpt11.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7		cntop2 14789	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 14606	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		cnf2 14792	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
12	1, 10, 6, 11	syl3anc 1250	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ 𝐾)
13	12	fvmptelcdm 5756	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐾)
14		eqid 2207	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
15	14	fvmpt2 5686	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐾) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
16	5, 13, 15	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
17		cnmpt1t.b	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
18		cntop2 14789	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
19	17, 18	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
20		toptopon2 14606	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
21	19, 20	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
22		cnf2 14792	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
23	1, 21, 17, 22	syl3anc 1250	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶∪ 𝐿)
24	23	fvmptelcdm 5756	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ∪ 𝐿)
25		eqid 2207	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
26	25	fvmpt2 5686	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
27	5, 24, 26	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
28	16, 27	opeq12d 3841	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
29	28	mpteq2dva 4150	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
30	4, 29	eqtr3d 2242	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
31		eqid 2207	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
32		nfcv 2350	. . . . 5 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩
33		nffvmpt1 5610	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦)
34		nffvmpt1 5610	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)
35	33, 34	nfop 3849	. . . . 5 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩
36		fveq2 5599	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦))
37		fveq2 5599	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦))
38	36, 37	opeq12d 3841	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑦)⟩)
39	32, 35, 38	cbvmpt 4155	. . . 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 14860	. . 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 411	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝐽 ↦ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
42	30, 41	eqeltrrd 2285	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2178  ⟨cop 3646  ∪ cuni 3864   ↦ cmpt 4121  ⟶wf 5286  ‘cfv 5290  (class class class)co 5967  Topctop 14584  TopOnctopon 14597   Cn ccn 14772   ×_t ctx 14839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-topgen 13207  df-top 14585  df-topon 14598  df-bases 14630  df-cn 14775  df-tx 14840

This theorem is referenced by:  cnmpt12f  14873  imasnopn  14886  cnrehmeocntop  15197