Theorem cnmpt21 23526

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 7407	. . . . . . . . . 10 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)
2		simprl 768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋)
3		simprr 770	. . . . . . . . . . 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 23446	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	4, 5, 6	syl2anc 583	. . . . . . . . . . . . . . 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 23104	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 8050	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
14	11, 13	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍)
15		rsp2 3268	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
16	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍))
17	16	imp 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍)
18	12	ovmpt4g 7550	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
19	2, 3, 17, 18	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
20	1, 19	eqtr3id 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
21	20	fveq2d 6888	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴))
22		eqid 2726	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)
23		cnmpt21.c	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
24	23	eleq1d 2812	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀))
25		cnmpt21.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
26		cntop2 23096	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ Top)
28		toptopon2 22771	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
29	27, 28	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
30		cnf2 23104	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
31	8, 29, 25, 30	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
32	22	fmpt 7104	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
33	31, 32	sylibr 233	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
35	24, 34, 17	rspcdva 3607	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀)
36	22, 23, 17, 35	fvmptd3 7014	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶)
37	21, 36	eqtrd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38		opelxpi 5706	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39		fvco3 6983	. . . . . . . 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 595	. . . . . . 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 7407	. . . . . . . 8 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
42		eqid 2726	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
43	42	ovmpt4g 7550	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
44	2, 3, 35, 43	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
45	41, 44	eqtr3id 2780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
46	37, 40, 45	3eqtr4d 2776	. . . . . 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 3194	. . . . 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 1909	. . . . . 6 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
49		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
50		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵)
51		nfmpo1 7484	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
52	50, 51	nfco 5858	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
53		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩
54	52, 53	nffv 6894	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩)
55		nfmpo1 7484	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
56	55, 53	nffv 6894	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
57	54, 56	nfeq 2910	. . . . . . 7 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
58	49, 57	nfralw 3302	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
59		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
60		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵)
61		nfmpo2 7485	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
62	60, 61	nfco 5858	. . . . . . . . . 10 ⊢ Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
63		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, 𝑣⟩
64	62, 63	nffv 6894	. . . . . . . . 9 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩)
65		nfmpo2 7485	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
66	65, 63	nffv 6894	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
67	64, 66	nfeq 2910	. . . . . . . 8 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
68		opeq2 4869	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
69	68	fveq2d 6888	. . . . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩))
71	69, 70	eqeq12d 2742	. . . . . . . 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 3297	. . . . . . 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 4868	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
74	73	fveq2d 6888	. . . . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
76	74, 75	eqeq12d 2742	. . . . . . . 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 3171	. . . . . . 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 283	. . . . . 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 3297	. . . . 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 217	. . . 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 6884	. . . . . 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 6884	. . . . . 6 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
83	81, 82	eqeq12d 2742	. . . . 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 5834	. . . 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 233	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤))
86		fco 6734	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀)
88	87	ffnd 6711	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌))
89	35	ralrimivva 3194	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀)
90	42	fmpo 8050	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
91	89, 90	sylib 217	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀)
92	91	ffnd 6711	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌))
93		eqfnfv 7025	. . . 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 583	. . 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 257	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
96		cnco 23121	. . 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 583	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
98	95, 97	eqeltrrd 2828	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ⟨cop 4629  ∪ cuni 4902   ↦ cmpt 5224   × cxp 5667   ∘ ccom 5673   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404   ∈ cmpo 7406  Topctop 22746  TopOnctopon 22763   Cn ccn 23079   ×_t ctx 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-topgen 17396  df-top 22747  df-topon 22764  df-bases 22800  df-cn 23082  df-tx 23417

This theorem is referenced by:  cnmpt21f  23527  xkofvcn  23539  xkohmeo  23670  qustgplem  23976  prdstmdd  23979  divcnOLD  24735  divcn  24737  htpycom  24853  htpycc  24857  reparphti  24874  reparphtiOLD  24875  pcocn  24895  pcohtpylem  24897  pcopt  24900  pcopt2  24901  pcoass  24902  pcorevlem  24904  dipcn  30478