Theorem cnmpt22 12502

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))
cnmpt22.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt22.m	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
cnmpt22.c	⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
cnmpt22.d	⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cnmpt22	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵   𝑤,𝐷,𝑧   𝑧,𝐽   𝑥,𝑤,𝑦,𝑧,𝐿   𝜑,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝑤,𝑀,𝑥,𝑦,𝑧   𝑤,𝑁,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑧,𝐾   𝑤,𝑊,𝑥,𝑦,𝑧   𝑤,𝑍,𝑥,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦

Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑧,𝑤)   𝐷(𝑥,𝑦)   𝐽(𝑥,𝑦,𝑤)   𝐾(𝑥,𝑦,𝑤)

Proof of Theorem cnmpt22

Step	Hyp	Ref	Expression
1		df-ov 5785	. . . 4 ⊢ (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)
2		cnmpt21.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		cnmpt21.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 12470	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		cnmpt22.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
7		cnmpt21.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
8		cnf2 12413	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
9	5, 6, 7, 8	syl3anc 1217	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
10		eqid 2140	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11	10	fmpo 6107	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12	9, 11	sylibr 133	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
13		rsp2 2485	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
14	12, 13	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
15	14	3impib 1180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
16		cnmpt22.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊))
17		cnmpt2t.b	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
18		cnf2 12413	. . . . . . . . 9 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘𝑊) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
19	5, 16, 17, 18	syl3anc 1217	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
20		eqid 2140	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	20	fmpo 6107	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
22	19, 21	sylibr 133	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊)
23		rsp2 2485	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
24	22, 23	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
25	24	3impib 1180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊)
26	15, 25	jca 304	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊))
27		txtopon 12470	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
28	6, 16, 27	syl2anc 409	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
29		cnmpt22.c	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
30		cntop2 12410	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
31	29, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Top)
32		toptopon2 12225	. . . . . . . . . . 11 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
33	31, 32	sylib 121	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
34		cnf2 12413	. . . . . . . . . 10 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
35	28, 33, 29, 34	syl3anc 1217	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
36		eqid 2140	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) = (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)
37	36	fmpo 6107	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
38	35, 37	sylibr 133	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁)
39		r2al 2457	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
40	38, 39	sylib 121	. . . . . . 7 ⊢ (𝜑 → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
41	40	3ad2ant1 1003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
42		eleq1 2203	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑍 ↔ 𝐴 ∈ 𝑍))
43		eleq1 2203	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
44	42, 43	bi2anan9 596	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → ((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) ↔ (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊)))
45		cnmpt22.d	. . . . . . . . 9 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)
46	45	eleq1d 2209	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝐶 ∈ ∪ 𝑁 ↔ 𝐷 ∈ ∪ 𝑁))
47	44, 46	imbi12d 233	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) ↔ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
48	47	spc2gv 2780	. . . . . 6 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) → ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
49	26, 41, 26, 48	syl3c 63	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ ∪ 𝑁)
50	45, 36	ovmpoga 5908	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊 ∧ 𝐷 ∈ ∪ 𝑁) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
51	15, 25, 49, 50	syl3anc 1217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
52	1, 51	syl5eqr 2187	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
53	52	mpoeq3dva 5843	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷))
54	2, 3, 7, 17	cnmpt2t 12501	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
55	2, 3, 54, 29	cnmpt21f 12500	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
56	53, 55	eqeltrrd 2218	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ⟨cop 3535  ∪ cuni 3744   × cxp 4545  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782   ∈ cmpo 5784  Topctop 12203  TopOnctopon 12216   Cn ccn 12393   ×_t ctx 12460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-topgen 12180  df-top 12204  df-topon 12217  df-bases 12249  df-cn 12396  df-tx 12461

This theorem is referenced by:  cnmpt22f  12503