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Theorem txcnp 22225

Description: If two functions are continuous at 𝐷, then the ordered pair of them is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypotheses**
Ref	Expression
txcnp.4	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txcnp.5	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txcnp.6	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
txcnp.7	⊢ (𝜑 → 𝐷 ∈ 𝑋)
txcnp.8	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
txcnp.9	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))

**Assertion**
Ref	Expression
txcnp	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×_t 𝐿))‘𝐷))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑌 𝑥,𝑍 𝑥,𝐷 𝑥,𝑋 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐽(𝑥) 𝐾(𝑥) 𝐿(𝑥)

Proof of Theorem txcnp Dummy variables 𝑠 𝑟 𝑡 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txcnp.4	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		txcnp.5	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		txcnp.8	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
4		cnpf2 21855	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
5	1, 2, 3, 4	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	5	fvmptelrn 6854	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
7		txcnp.6	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
8		txcnp.9	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
9		cnpf2 21855	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
10	1, 7, 8, 9	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
11	10	fvmptelrn 6854	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍)
12	6, 11	opelxpd 5557	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
13	12	fmpttd 6856	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍))
14		txcnp.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
15		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
16		opex 5321	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
17		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
18	17	fvmpt2 6756	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
19	15, 16, 18	sylancl 589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
20		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
21	20	fvmpt2 6756	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
22	15, 6, 21	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
23		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
24	23	fvmpt2 6756	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ 𝑍) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
25	15, 11, 24	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
26	22, 25	opeq12d 4773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
27	19, 26	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
28	27	ralrimiva 3149	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
29		nffvmpt1 6656	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷)
30		nffvmpt1 6656	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)
31		nffvmpt1 6656	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)
32	30, 31	nfop 4781	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩
33	29, 32	nfeq 2968	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩
34		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷))
35		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷))
36		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷))
37	35, 36	opeq12d 4773	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩)
38	34, 37	eqeq12d 2814	. . . . . . . . . 10 ⊢ (𝑥 = 𝐷 → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩))
39	33, 38	rspc 3559	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩))
40	14, 28, 39	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩)
41	40	eleq1d 2874	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
42	41	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
43	3	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
44		simplrl 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑣 ∈ 𝐾)
45		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣)
46		cnpimaex 21861	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣 ∈ 𝐾 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣))
47	43, 44, 45, 46	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣))
48	8	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
49		simplrr 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑤 ∈ 𝐿)
50		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)
51		cnpimaex 21861	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤 ∈ 𝐿 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))
52	48, 49, 50, 51	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))
53	47, 52	jca 515	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))
54	53	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) → (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))))
55		opelxp 5555	. . . . . . 7 ⊢ (⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) ↔ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤))
56		reeanv 3320	. . . . . . 7 ⊢ (∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) ↔ (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))
57	54, 55, 56	3imtr4g 299	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))))
58	42, 57	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))))
59		an4 655	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) ↔ ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))
60		elin 3897	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (𝑟 ∩ 𝑠) ↔ (𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠))
61	60	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠))
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠)))
63		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → 𝑟 ∈ 𝐽)
64		toponss 21532	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝐽) → 𝑟 ⊆ 𝑋)
65	1, 63, 64	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → 𝑟 ⊆ 𝑋)
66		ssinss1 4164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ⊆ 𝑋 → (𝑟 ∩ 𝑠) ⊆ 𝑋)
67	66	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ 𝑋)
68	67	sselda 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑋)
69	28	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
70		nffvmpt1 6656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡)
71		nffvmpt1 6656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡)
72		nffvmpt1 6656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)
73	71, 72	nfop 4781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩
74	70, 73	nfeq 2968	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩
75		fveq2 6645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡))
76		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡))
77		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡))
78	76, 77	opeq12d 4773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑡 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩)
79	75, 78	eqeq12d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑡 → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩))
80	74, 79	rspc 3559	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩))
81	68, 69, 80	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩)
82		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ (𝑟 ∩ 𝑠))
83	82	elin1d 4125	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑟)
84	5	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
85	84	ffund 6491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐴))
86	67	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ 𝑋)
87	84	fdmd 6497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋)
88	86, 87	sseqtrrd 3956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐴))
89	88, 82	sseldd 3916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴))
90		funfvima 6970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟)))
91	85, 89, 90	syl2anc 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟)))
92	83, 91	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟))
93	82	elin2d 4126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑠)
94	10	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
95	94	ffund 6491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐵))
96	94	fdmd 6497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
97	86, 96	sseqtrrd 3956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐵))
98	97, 82	sseldd 3916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵))
99		funfvima 6970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ 𝐵) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
100	95, 98, 99	syl2anc 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
101	93, 100	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))
102	92, 101	opelxpd 5557	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩ ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
103	81, 102	eqeltrd 2890	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
104	103	ralrimiva 3149	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
105	13	ffund 6491	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
106	105	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → Fun (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
107	13	fdmd 6497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
108	107	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
109	67, 108	sseqtrrd 3956	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
110		funimass4 6705	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∧ (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))))
111	106, 109, 110	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))))
112	104, 111	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
113	65, 112	syldan 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
114	113	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
115		xpss12 5534	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤))
116		sstr2 3922	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))
117	114, 115, 116	syl2im 40	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))
118	62, 117	anim12d 611	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))
119	59, 118	syl5bi 245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))
120		topontop 21518	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
121	1, 120	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
122		inopn 21504	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (𝑟 ∩ 𝑠) ∈ 𝐽)
123	122	3expb 1117	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
124	121, 123	sylan 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
125	124	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
126	119, 125	jctild 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))))
127	126	expimpd 457	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) ∧ ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))))
128		eleq2 2878	. . . . . . . . . 10 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → (𝐷 ∈ 𝑧 ↔ 𝐷 ∈ (𝑟 ∩ 𝑠)))
129		imaeq2 5892	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)))
130	129	sseq1d 3946	. . . . . . . . . 10 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))
131	128, 130	anbi12d 633	. . . . . . . . 9 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))
132	131	rspcev 3571	. . . . . . . 8 ⊢ (((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
133	127, 132	syl6 35	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) ∧ ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
134	133	expd 419	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → ((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
135	134	rexlimdvv 3252	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
136	58, 135	syld 47	. . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
137	136	ralrimivva 3156	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
138		vex 3444	. . . . . 6 ⊢ 𝑣 ∈ V
139		vex 3444	. . . . . 6 ⊢ 𝑤 ∈ V
140	138, 139	xpex 7456	. . . . 5 ⊢ (𝑣 × 𝑤) ∈ V
141	140	rgen2w 3119	. . . 4 ⊢ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (𝑣 × 𝑤) ∈ V
142		eqid 2798	. . . . 5 ⊢ (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))
143		eleq2 2878	. . . . . 6 ⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤)))
144		sseq2 3941	. . . . . . . 8 ⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
145	144	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = (𝑣 × 𝑤) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
146	145	rexbidv 3256	. . . . . 6 ⊢ (𝑦 = (𝑣 × 𝑤) → (∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
147	143, 146	imbi12d 348	. . . . 5 ⊢ (𝑦 = (𝑣 × 𝑤) → ((((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
148	142, 147	ralrnmpo 7268	. . . 4 ⊢ (∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (𝑣 × 𝑤) ∈ V → (∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
149	141, 148	ax-mp 5	. . 3 ⊢ (∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
150	137, 149	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))
151		topontop 21518	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
152	2, 151	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
153		topontop 21518	. . . . 5 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
154	7, 153	syl 17	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Top)
155		eqid 2798	. . . . 5 ⊢ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))
156	155	txval 22169	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×_t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))))
157	152, 154, 156	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐾 ×_t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))))
158		txtopon 22196	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×_t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
159	2, 7, 158	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐾 ×_t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
160	1, 157, 159, 14	tgcnp 21858	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×_t 𝐿))‘𝐷) ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) ∧ ∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))))
161	13, 150, 160	mpbir2and 712	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×_t 𝐿))‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ⟨cop 4531 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 “ cima 5522 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 topGenctg 16703 Topctop 21498 TopOnctopon 21515 CnP ccnp 21830 ×_t ctx 22165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cnp 21833 df-tx 22167

This theorem is referenced by: limccnp2 24495

W3C validator