Theorem cnmpt21 24045

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 7454	. . . . . . . . . 10 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)
2		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
3		simprr 785	. . . . . . . . . . 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 23965	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	4, 5, 6	syl2anc 596	. . . . . . . . . . . . . . 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 23622	. . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 8114	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
14	11, 13	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
15		rsp2 3283	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
16	14, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
17	16	imp 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍)
18	12	ovmpt4g 7599	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
19	2, 3, 17, 18	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
20	1, 19	eqtr3id 2812	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
21	20	fveq2d 6920	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴))
22		eqid 2763	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)
23		cnmpt21.c	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
24	23	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀))
25		cnmpt21.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
26		cntop2 23614	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ Top)
28		toptopon2 23291	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
29	27, 28	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
30		cnf2 23622	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
31	8, 29, 25, 30	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
32	22	fmpt 7142	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
33	31, 32	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
34	33	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
35	24, 34, 17	rspcdva 3581	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀)
36	22, 23, 17, 35	fvmptd3 7048	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶)
37	21, 36	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38		opelxpi 5695	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39		fvco3 7016	. . . . . . . 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 608	. . . . . . 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 7454	. . . . . . . 8 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
42		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
43	42	ovmpt4g 7599	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
44	2, 3, 35, 43	syl3anc 1399	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
45	41, 44	eqtr3id 2812	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
46	37, 40, 45	3eqtr4d 2808	. . . . . 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 3209	. . . . 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 1949	. . . . . 6 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
49		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
50		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵)
51		nfmpo1 7531	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
52	50, 51	nfco 5854	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
53		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩
54	52, 53	nffv 6926	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩)
55		nfmpo1 7531	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
56	55, 53	nffv 6926	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
57	54, 56	nfeq 2939	. . . . . . 7 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
58	49, 57	nfralw 3313	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
59		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
60		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵)
61		nfmpo2 7532	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
62	60, 61	nfco 5854	. . . . . . . . . 10 ⊢ Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
63		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, 𝑣⟩
64	62, 63	nffv 6926	. . . . . . . . 9 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩)
65		nfmpo2 7532	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
66	65, 63	nffv 6926	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
67	64, 66	nfeq 2939	. . . . . . . 8 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
68		opeq2 4846	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
69	68	fveq2d 6920	. . . . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩))
71	69, 70	eqeq12d 2779	. . . . . . . 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	cbvralw 3308	. . . . . . 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 4845	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
74	73	fveq2d 6920	. . . . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
76	74, 75	eqeq12d 2779	. . . . . . . 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 3189	. . . . . . 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 286	. . . . . 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	cbvralw 3308	. . . . 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 221	. . . 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 6916	. . . . . 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 6916	. . . . . 6 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
83	81, 82	eqeq12d 2779	. . . . 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 5828	. . . 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 237	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤))
86		fco 6764	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀)
88	87	ffnd 6740	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌))
89	35	ralrimivva 3209	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀)
90	42	fmpo 8114	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
91	89, 90	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
92	91	ffnd 6740	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌))
93		eqfnfv 7060	. . . 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 596	. . 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 260	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
96		cnco 23639	. . 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 596	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
98	95, 97	eqeltrrd 2865	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4840  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5656   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Topctop 23266  TopOnctopon 23283   Cn ccn 23597   ×_t ctx 23934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-topgen 17664  df-top 23267  df-topon 23284  df-bases 23319  df-cn 23600  df-tx 23936

This theorem is used by:  cnmpt21f  24046  xkofvcn  24058  xkohmeo  24189  qustgplem  24495  prdstmdd  24498  divcn  25244  htpycom  25352  htpycc  25356  reparphti  25373  pcocn  25393  pcohtpylem  25395  pcopt  25398  pcopt2  25399  pcoass  25400  pcorevlem  25402  dipcn  31379