Theorem cnmpt21 14763

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))
cnmpt21.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt21.b	⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
cnmpt21.c	⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cnmpt21	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

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

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

Proof of Theorem cnmpt21

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 5947	. . . . . . . . . 10 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)
2		simprl 529	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
3		simprr 531	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌)
4		cnmpt21.j	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
5		cnmpt21.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
6		txtopon 14734	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	4, 5, 6	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
8		cnmpt21.l	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
9		cnmpt21.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
10		cnf2 14677	. . . . . . . . . . . . . . 15 ⊢ (((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)
11	7, 8, 9, 10	syl3anc 1250	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12		eqid 2205	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 6287	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
14	11, 13	sylibr 134	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
15		rsp2 2556	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
16	14, 15	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
17	16	imp 124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍)
18	12	ovmpt4g 6068	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
19	2, 3, 17, 18	syl3anc 1250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
20	1, 19	eqtr3id 2252	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
21	20	fveq2d 5580	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴))
22		eqid 2205	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)
23		cnmpt21.c	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
24	23	eleq1d 2274	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀))
25		cnmpt21.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
26		cntop2 14674	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
27	25, 26	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ Top)
28		toptopon2 14491	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
29	27, 28	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
30		cnf2 14677	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
31	8, 29, 25, 30	syl3anc 1250	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
32	22	fmpt 5730	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
33	31, 32	sylibr 134	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
34	33	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
35	24, 34, 17	rspcdva 2882	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀)
36	22, 23, 17, 35	fvmptd3 5673	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶)
37	21, 36	eqtrd 2238	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38		opelxpi 4707	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39		fvco3 5650	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)))
40	11, 38, 39	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)))
41		df-ov 5947	. . . . . . . 8 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
42		eqid 2205	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
43	42	ovmpt4g 6068	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
44	2, 3, 35, 43	syl3anc 1250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
45	41, 44	eqtr3id 2252	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
46	37, 40, 45	3eqtr4d 2248	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩))
47	46	ralrimivva 2588	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩))
48		nfv 1551	. . . . . 6 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
49		nfcv 2348	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
50		nfcv 2348	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵)
51		nfmpo1 6012	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
52	50, 51	nfco 4843	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
53		nfcv 2348	. . . . . . . . 9 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩
54	52, 53	nffv 5586	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩)
55		nfmpo1 6012	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
56	55, 53	nffv 5586	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
57	54, 56	nfeq 2356	. . . . . . 7 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
58	49, 57	nfralxy 2544	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
59		nfv 1551	. . . . . . . 8 ⊢ Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
60		nfcv 2348	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵)
61		nfmpo2 6013	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
62	60, 61	nfco 4843	. . . . . . . . . 10 ⊢ Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
63		nfcv 2348	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, 𝑣⟩
64	62, 63	nffv 5586	. . . . . . . . 9 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩)
65		nfmpo2 6013	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
66	65, 63	nffv 5586	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
67	64, 66	nfeq 2356	. . . . . . . 8 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
68		opeq2 3820	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
69	68	fveq2d 5580	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩))
70	68	fveq2d 5580	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩))
71	69, 70	eqeq12d 2220	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)))
72	59, 67, 71	cbvral 2734	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩))
73		opeq1 3819	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
74	73	fveq2d 5580	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩))
75	73	fveq2d 5580	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
76	74, 75	eqeq12d 2220	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)))
77	76	ralbidv 2506	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)))
78	72, 77	bitrid 192	. . . . . 6 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)))
79	48, 58, 78	cbvral 2734	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
80	47, 79	sylib 122	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
81		fveq2 5576	. . . . . 6 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩))
82		fveq2 5576	. . . . . 6 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
83	81, 82	eqeq12d 2220	. . . . 5 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)))
84	83	ralxp 4821	. . . 4 ⊢ (∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
85	80, 84	sylibr 134	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤))
86		fco 5441	. . . . . 6 ⊢ (((𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀 ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀)
87	31, 11, 86	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀)
88	87	ffnd 5426	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌))
89	35	ralrimivva 2588	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀)
90	42	fmpo 6287	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
91	89, 90	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
92	91	ffnd 5426	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌))
93		eqfnfv 5677	. . . 4 ⊢ ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤)))
94	88, 92, 93	syl2anc 411	. . 3 ⊢ (𝜑 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤)))
95	85, 94	mpbird 167	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
96		cnco 14693	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
97	9, 25, 96	syl2anc 411	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
98	95, 97	eqeltrrd 2283	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ⟨cop 3636  ∪ cuni 3850   ↦ cmpt 4105   × cxp 4673   ∘ ccom 4679   Fn wfn 5266  ⟶wf 5267  ‘cfv 5271  (class class class)co 5944   ∈ cmpo 5946  Topctop 14469  TopOnctopon 14482   Cn ccn 14657   ×_t ctx 14724

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-topgen 13092  df-top 14470  df-topon 14483  df-bases 14515  df-cn 14660  df-tx 14725

This theorem is referenced by:  cnmpt21f  14764  divcnap  15037