Theorem cnmpt21 22822

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 7278	. . . . . . . . . 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 22742	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7	4, 5, 6	syl2anc 584	. . . . . . . . . . . . . . 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 22400	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍)
12		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 7908	. . . . . . . . . . . . . 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 3138	. . . . . . . . . . . . 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 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍)
18	12	ovmpt4g 7420	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
19	2, 3, 17, 18	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
20	1, 19	eqtr3id 2792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
21	20	fveq2d 6778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴))
22		eqid 2738	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵)
23		cnmpt21.c	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶)
24	23	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀))
25		cnmpt21.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀))
26		cntop2 22392	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ Top)
28		toptopon2 22067	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
29	27, 28	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
30		cnf2 22400	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
31	8, 29, 25, 30	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
32	22	fmpt 6984	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀)
33	31, 32	sylibr 233	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
34	33	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀)
35	24, 34, 17	rspcdva 3562	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀)
36	22, 23, 17, 35	fvmptd3 6898	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶)
37	21, 36	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38		opelxpi 5626	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39		fvco3 6867	. . . . . . . 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 596	. . . . . . 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 7278	. . . . . . . 8 ⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
42		eqid 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
43	42	ovmpt4g 7420	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
44	2, 3, 35, 43	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶)
45	41, 44	eqtr3id 2792	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
46	37, 40, 45	3eqtr4d 2788	. . . . . 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 3123	. . . . 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 1917	. . . . . 6 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
49		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
50		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵)
51		nfmpo1 7355	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
52	50, 51	nfco 5774	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
53		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩
54	52, 53	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩)
55		nfmpo1 7355	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
56	55, 53	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
57	54, 56	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
58	49, 57	nfralw 3151	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩)
59		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩)
60		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵)
61		nfmpo2 7356	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
62	60, 61	nfco 5774	. . . . . . . . . 10 ⊢ Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
63		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, 𝑣⟩
64	62, 63	nffv 6784	. . . . . . . . 9 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩)
65		nfmpo2 7356	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
66	65, 63	nffv 6784	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
67	64, 66	nfeq 2920	. . . . . . . 8 ⊢ Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩)
68		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
69	68	fveq2d 6778	. . . . . . . . 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 6778	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩))
71	69, 70	eqeq12d 2754	. . . . . . . 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 3373	. . . . . . 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 4804	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
74	73	fveq2d 6778	. . . . . . . . 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 6778	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
76	74, 75	eqeq12d 2754	. . . . . . . 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 3112	. . . . . . 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 282	. . . . . 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 3373	. . . . 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 6774	. . . . . 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 6774	. . . . . 6 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘⟨𝑢, 𝑣⟩))
83	81, 82	eqeq12d 2754	. . . . 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 5750	. . . 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 6624	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀)
88	87	ffnd 6601	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌))
89	35	ralrimivva 3123	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀)
90	42	fmpo 7908	. . . . . 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 6601	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌))
93		eqfnfv 6909	. . . 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 584	. . 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 256	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
96		cnco 22417	. . 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 584	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
98	95, 97	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567  ∪ cuni 4839   ↦ cmpt 5157   × cxp 5587   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-tx 22713

This theorem is referenced by:  cnmpt21f  22823  xkofvcn  22835  xkohmeo  22966  qustgplem  23272  prdstmdd  23275  divcn  24031  htpycom  24139  htpycc  24143  reparphti  24160  pcocn  24180  pcohtpylem  24182  pcopt  24185  pcopt2  24186  pcoass  24187  pcorevlem  24189  dipcn  29082