Theorem cnmpt22 21477

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 6653	. . . 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 21394	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 693	. . . . . . . . 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 21053	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
10		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11	10	fmpt2 7237	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12	9, 11	sylibr 224	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
13		rsp2 2936	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
15	14	3impib 1262	. . . . 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 21053	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
20		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	20	fmpt2 7237	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶𝑊)
22	19, 21	sylibr 224	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊)
23		rsp2 2936	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ 𝑊 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊))
25	24	3impib 1262	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ 𝑊)
26	15, 25	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊))
27		txtopon 21394	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑊)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑍 × 𝑊)))
28	6, 16, 27	syl2anc 693	. . . . . . . . . 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 21045	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ Top)
32		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ 𝑁 = ∪ 𝑁
33	32	toptopon 20722	. . . . . . . . . . 11 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
34	31, 33	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
35		cnf2 21053	. . . . . . . . . 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)
36	28, 34, 29, 35	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
37		eqid 2622	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) = (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)
38	37	fmpt2 7237	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶):(𝑍 × 𝑊)⟶∪ 𝑁)
39	36, 38	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁)
40		r2al 2939	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑍 ∀𝑤 ∈ 𝑊 𝐶 ∈ ∪ 𝑁 ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
41	39, 40	sylib 208	. . . . . . 7 ⊢ (𝜑 → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
42	41	3ad2ant1 1082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁))
43		eleq1 2689	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑍 ↔ 𝐴 ∈ 𝑍))
44		eleq1 2689	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (𝑤 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
45	43, 44	bi2anan9 917	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → ((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) ↔ (𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊)))
46		cnmpt22.d	. . . . . . . . 9 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷)
47	46	eleq1d 2686	. . . . . . . 8 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝐶 ∈ ∪ 𝑁 ↔ 𝐷 ∈ ∪ 𝑁))
48	45, 47	imbi12d 334	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) ↔ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
49	48	spc2gv 3296	. . . . . 6 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → (∀𝑧∀𝑤((𝑧 ∈ 𝑍 ∧ 𝑤 ∈ 𝑊) → 𝐶 ∈ ∪ 𝑁) → ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊) → 𝐷 ∈ ∪ 𝑁)))
50	26, 42, 26, 49	syl3c 66	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ ∪ 𝑁)
51	46, 37	ovmpt2ga 6790	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑊 ∧ 𝐷 ∈ ∪ 𝑁) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
52	15, 25, 50, 51	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴(𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)𝐵) = 𝐷)
53	1, 52	syl5eqr 2670	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩) = 𝐷)
54	53	mpt2eq3dva 6719	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷))
55	2, 3, 7, 17	cnmpt2t 21476	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
56	2, 3, 55, 29	cnmpt21f 21475	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶)‘⟨𝐴, 𝐵⟩)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))
57	54, 56	eqeltrrd 2702	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-tx 21365

This theorem is referenced by:  cnmpt22f  21478  xkofvcn  21487  cnmptk2  21489  pcorevlem  22826