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Theorem txcnp 21634

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	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		txcnp.5	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		txcnp.8	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
4		cnpf2 21265	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
5	1, 2, 3, 4	syl3anc 1483	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6		eqid 2806	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
7	6	fmpt 6598	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
8	5, 7	sylibr 225	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
9	8	r19.21bi 3120	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
10		txcnp.6	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
11		txcnp.9	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
12		cnpf2 21265	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
13	1, 10, 11, 12	syl3anc 1483	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
14		eqid 2806	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
15	14	fmpt 6598	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
16	13, 15	sylibr 225	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑍)
17	16	r19.21bi 3120	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍)
18		opelxpi 5348	. . . 4 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
19	9, 17, 18	syl2anc 575	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
20	19	fmpttd 6603	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍))
21		txcnp.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
22		simpr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23		opex 5122	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
24		eqid 2806	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
25	24	fvmpt2 6508	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
26	22, 23, 25	sylancl 576	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
27	6	fvmpt2 6508	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
28	22, 9, 27	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
29	14	fvmpt2 6508	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ 𝑍) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
30	22, 17, 29	syl2anc 575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵)
31	28, 30	opeq12d 4603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
32	26, 31	eqtr4d 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
33	32	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
34		nffvmpt1 6415	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷)
35		nffvmpt1 6415	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)
36		nffvmpt1 6415	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)
37	35, 36	nfop 4611	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩
38	34, 37	nfeq 2960	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩
39		fveq2 6404	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷))
40		fveq2 6404	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷))
41		fveq2 6404	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷))
42	40, 41	opeq12d 4603	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐷 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩)
43	39, 42	eqeq12d 2821	. . . . . . . . . 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)⟩))
44	38, 43	rspc 3496	. . . . . . . . 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)⟩))
45	21, 33, 44	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩)
46	45	eleq1d 2870	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
47	46	adantr 468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
48	3	ad2antrr 708	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
49		simplrl 786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑣 ∈ 𝐾)
50		simprl 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣)
51		cnpimaex 21271	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣 ∈ 𝐾 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣))
52	48, 49, 50, 51	syl3anc 1483	. . . . . . . . 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))
53	11	ad2antrr 708	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
54		simplrr 787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑤 ∈ 𝐿)
55		simprr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)
56		cnpimaex 21271	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤 ∈ 𝐿 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))
57	53, 54, 55, 56	syl3anc 1483	. . . . . . . . 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))
58	52, 57	jca 503	. . . . . . . 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)))
59	58	ex 399	. . . . . . 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))))
60		opelxp 5346	. . . . . . 7 ⊢ (⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) ↔ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤))
61		reeanv 3295	. . . . . . 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)))
62	59, 60, 61	3imtr4g 287	. . . . . 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))))
63	47, 62	sylbid 231	. . . . 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))))
64		an4 638	. . . . . . . . . . 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)))
65		elin 3995	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (𝑟 ∩ 𝑠) ↔ (𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠))
66	65	biimpri 219	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠))
67	66	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠)))
68		simpl 470	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → 𝑟 ∈ 𝐽)
69		toponss 20942	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝐽) → 𝑟 ⊆ 𝑋)
70	1, 68, 69	syl2an 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → 𝑟 ⊆ 𝑋)
71		ssinss1 4038	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ⊆ 𝑋 → (𝑟 ∩ 𝑠) ⊆ 𝑋)
72	71	adantl 469	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ 𝑋)
73	72	sselda 3798	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑋)
74	33	ad2antrr 708	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩)
75		nffvmpt1 6415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡)
76		nffvmpt1 6415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡)
77		nffvmpt1 6415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)
78	76, 77	nfop 4611	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩
79	75, 78	nfeq 2960	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩
80		fveq2 6404	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡))
81		fveq2 6404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡))
82		fveq2 6404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡))
83	81, 82	opeq12d 4603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑡 → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩)
84	80, 83	eqeq12d 2821	. . . . . . . . . . . . . . . . . . . 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)⟩))
85	79, 84	rspc 3496	. . . . . . . . . . . . . . . . . . 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)⟩))
86	73, 74, 85	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩)
87		inss1 4029	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∩ 𝑠) ⊆ 𝑟
88		simpr 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ (𝑟 ∩ 𝑠))
89	87, 88	sseldi 3796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑟)
90	5	ad2antrr 708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
91	90	ffund 6256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐴))
92	72	adantr 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ 𝑋)
93	90	fdmd 6261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋)
94	92, 93	sseqtr4d 3839	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐴))
95	94, 88	sseldd 3799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴))
96		funfvima 6713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟)))
97	91, 95, 96	syl2anc 575	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟)))
98	89, 97	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟))
99		inss2 4030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∩ 𝑠) ⊆ 𝑠
100	99, 88	sseldi 3796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑠)
101	13	ad2antrr 708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
102	101	ffund 6256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐵))
103	101	fdmd 6261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
104	92, 103	sseqtr4d 3839	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐵))
105	104, 88	sseldd 3799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵))
106		funfvima 6713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ 𝐵) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
107	102, 105, 106	syl2anc 575	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
108	100, 107	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))
109		opelxpi 5348	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩ ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
110	98, 108, 109	syl2anc 575	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ⟨((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)⟩ ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
111	86, 110	eqeltrd 2885	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
112	111	ralrimiva 3154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
113	20	ffund 6256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
114	113	adantr 468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → Fun (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
115	20	fdmd 6261	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
116	115	adantr 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
117	72, 116	sseqtr4d 3839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
118		funimass4 6464	. . . . . . . . . . . . . . . . 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))))
119	114, 117, 118	syl2anc 575	. . . . . . . . . . . . . . . 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))))
120	112, 119	mpbird 248	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
121	70, 120	syldan 581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
122	121	adantlr 697	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))
123		xpss12 5326	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤))
124		sstr2 3805	. . . . . . . . . . . . 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)))
125	122, 123, 124	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)))
126	67, 125	anim12d 598	. . . . . . . . . . 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))))
127	64, 126	syl5bi 233	. . . . . . . . . 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))))
128		topontop 20928	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
129	1, 128	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
130		inopn 20914	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (𝑟 ∩ 𝑠) ∈ 𝐽)
131	130	3expb 1142	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
132	129, 131	sylan 571	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
133	132	adantlr 697	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽)
134	127, 133	jctild 517	. . . . . . . . 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)))))
135	134	expimpd 443	. . . . . . . 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)))))
136		eleq2 2874	. . . . . . . . . 10 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → (𝐷 ∈ 𝑧 ↔ 𝐷 ∈ (𝑟 ∩ 𝑠)))
137		imaeq2 5672	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) = ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)))
138	137	sseq1d 3829	. . . . . . . . . 10 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))
139	136, 138	anbi12d 618	. . . . . . . . 9 ⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))
140	139	rspcev 3502	. . . . . . . 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)))
141	135, 140	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))))
142	141	expd 402	. . . . . 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)))))
143	142	rexlimdvv 3225	. . . . 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))))
144	63, 143	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))))
145	144	ralrimivva 3159	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
146		vex 3394	. . . . . 6 ⊢ 𝑣 ∈ V
147		vex 3394	. . . . . 6 ⊢ 𝑤 ∈ V
148	146, 147	xpex 7188	. . . . 5 ⊢ (𝑣 × 𝑤) ∈ V
149	148	rgen2w 3113	. . . 4 ⊢ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (𝑣 × 𝑤) ∈ V
150		eqid 2806	. . . . 5 ⊢ (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))
151		eleq2 2874	. . . . . 6 ⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤)))
152		sseq2 3824	. . . . . . . 8 ⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
153	152	anbi2d 616	. . . . . . 7 ⊢ (𝑦 = (𝑣 × 𝑤) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
154	153	rexbidv 3240	. . . . . 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))))
155	151, 154	imbi12d 335	. . . . 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)))))
156	150, 155	ralrnmpt2 7001	. . . 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)))))
157	149, 156	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))))
158	145, 157	sylibr 225	. 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)))
159		topontop 20928	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
160	2, 159	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
161		topontop 20928	. . . . 5 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
162	10, 161	syl 17	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Top)
163		eqid 2806	. . . . 5 ⊢ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))
164	163	txval 21578	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×_t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))))
165	160, 162, 164	syl2anc 575	. . 3 ⊢ (𝜑 → (𝐾 ×_t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))))
166		txtopon 21605	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×_t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
167	2, 10, 166	syl2anc 575	. . 3 ⊢ (𝜑 → (𝐾 ×_t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
168	1, 165, 167, 21	tgcnp 21268	. 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)))))
169	20, 158, 168	mpbir2and 695	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×_t 𝐿))‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 197 ∧ wa 384 = wceq 1637 ∈ wcel 2156 ∀wral 3096 ∃wrex 3097 Vcvv 3391 ∩ cin 3768 ⊆ wss 3769 ⟨cop 4376 ↦ cmpt 4923 × cxp 5309 dom cdm 5311 ran crn 5312 “ cima 5314 Fun wfun 6091 ⟶wf 6093 ‘cfv 6097 (class class class)co 6870 ↦ cmpt2 6872 topGenctg 16299 Topctop 20908 TopOnctopon 20925 CnP ccnp 21240 ×_t ctx 21574

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7175

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-ral 3101 df-rex 3102 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-op 4377 df-uni 4631 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5219 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6060 df-fun 6099 df-fn 6100 df-f 6101 df-fv 6105 df-ov 6873 df-oprab 6874 df-mpt2 6875 df-1st 7394 df-2nd 7395 df-map 8090 df-topgen 16305 df-top 20909 df-topon 20926 df-bases 20961 df-cnp 21243 df-tx 21576

This theorem is referenced by: limccnp2 23869

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