Theorem txrest 14444

Description: The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
txrest	⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)))

Proof of Theorem txrest

Dummy variables 𝑠 𝑟 𝑢 𝑣 𝑥 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . . 6 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
2	1	txval 14423	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
3	2	adantr 276	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
4	3	oveq1d 5933	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
5	1	txbasex 14425	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6		xpexg 4773	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 × 𝐵) ∈ V)
7		tgrest 14337	. . . 4 ⊢ ((ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
8	5, 6, 7	syl2an 289	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9		elrest 12857	. . . . . . . 8 ⊢ ((ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
10	5, 6, 9	syl2an 289	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11		vex 2763	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
12	11	inex1 4163	. . . . . . . . . 10 ⊢ (𝑟 ∩ 𝐴) ∈ V
13	12	a1i 9	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ∩ 𝐴) ∈ V)
14		elrest 12857	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
15	14	ad2ant2r 509	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
16		xpeq1 4673	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑢 × 𝑣) = ((𝑟 ∩ 𝐴) × 𝑣))
17	16	eqeq2d 2205	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
18	17	rexbidv 2495	. . . . . . . . . 10 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
19		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
20	19	inex1 4163	. . . . . . . . . . . 12 ⊢ (𝑠 ∩ 𝐵) ∈ V
21	20	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝐵) ∈ V)
22		elrest 12857	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐵 ∈ 𝑌) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
23	22	ad2ant2l 508	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
24		xpeq2 4674	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → ((𝑟 ∩ 𝐴) × 𝑣) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
25	24	eqeq2d 2205	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
26	25	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑣 = (𝑠 ∩ 𝐵)) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
27	21, 23, 26	rexxfr2d 4496	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
28	18, 27	sylan9bbr 463	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑢 = (𝑟 ∩ 𝐴)) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
29	13, 15, 28	rexxfr2d 4496	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
30	11, 19	xpex 4774	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
31	30	rgen2w 2550	. . . . . . . . 9 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
32		eqid 2193	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
33		ineq1 3353	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34		inxp 4796	. . . . . . . . . . . 12 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
35	33, 34	eqtrdi 2242	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
36	35	eqeq2d 2205	. . . . . . . . . 10 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
37	32, 36	rexrnmpo 6034	. . . . . . . . 9 ⊢ (∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V → (∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
38	31, 37	ax-mp 5	. . . . . . . 8 ⊢ (∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
39	29, 38	bitr4di 198	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
40	10, 39	bitr4d 191	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑥 ∈ (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣)))
41	40	abbi2dv 2312	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣)})
42		eqid 2193	. . . . . 6 ⊢ (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
43	42	rnmpo 6029	. . . . 5 ⊢ ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅 ↾t 𝐴)∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣)}
44	41, 43	eqtr4di 2244	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))
45	44	fveq2d 5558	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (topGen‘(ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))))
46	4, 8, 45	3eqtr2d 2232	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))))
47		restfn 12854	. . . 4 ⊢ ↾t Fn (V × V)
48		simpll 527	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑅 ∈ 𝑉)
49	48	elexd 2773	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑅 ∈ V)
50		simprl 529	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
51	50	elexd 2773	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ V)
52		fnovex 5951	. . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝑅 ∈ V ∧ 𝐴 ∈ V) → (𝑅 ↾t 𝐴) ∈ V)
53	47, 49, 51, 52	mp3an2i 1353	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑅 ↾t 𝐴) ∈ V)
54		simplr 528	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑆 ∈ 𝑊)
55	54	elexd 2773	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑆 ∈ V)
56		simprr 531	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
57	56	elexd 2773	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ V)
58		fnovex 5951	. . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝑆 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ↾t 𝐵) ∈ V)
59	47, 55, 57, 58	mp3an2i 1353	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑆 ↾t 𝐵) ∈ V)
60		eqid 2193	. . . 4 ⊢ ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
61	60	txval 14423	. . 3 ⊢ (((𝑅 ↾t 𝐴) ∈ V ∧ (𝑆 ↾t 𝐵) ∈ V) → ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))))
62	53, 59, 61	syl2anc 411	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))))
63	46, 62	eqtr4d 2229	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ∩ cin 3152   × cxp 4657  ran crn 4660   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920   ↾_t crest 12850  topGenctg 12865   ×_t ctx 14420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-rest 12852  df-topgen 12871  df-tx 14421

This theorem is referenced by:  cnmpt2res  14465  limccnp2cntop  14831