Theorem txrest 23977

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 2763	. . . . . 6 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
2	1	txval 23910	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
3	2	adantr 486	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
4	3	oveq1d 7455	. . 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 23912	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6		xpexg 7785	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 × 𝐵) ∈ V)
7		tgrest 23504	. . . 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 608	. . 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 17624	. . . . . . . 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 608	. . . . . . 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 3458	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
12	11	inex1 5281	. . . . . . . . . 10 ⊢ (𝑟 ∩ 𝐴) ∈ V
13	12	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ∩ 𝐴) ∈ V)
14		elrest 17624	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
15	14	ad2ant2r 760	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
16		xpeq1 5669	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑢 × 𝑣) = ((𝑟 ∩ 𝐴) × 𝑣))
17	16	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
18	17	rexbidv 3190	. . . . . . . . . 10 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
19		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
20	19	inex1 5281	. . . . . . . . . . . 12 ⊢ (𝑠 ∩ 𝐵) ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝐵) ∈ V)
22		elrest 17624	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐵 ∈ 𝑌) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
23	22	ad2ant2l 759	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
24		xpeq2 5676	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → ((𝑟 ∩ 𝐴) × 𝑣) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
25	24	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
26	25	adantl 487	. . . . . . . . . . 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 5377	. . . . . . . . . 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 520	. . . . . . . . 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 5377	. . . . . . . 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 7788	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
31	30	rgen2w 3085	. . . . . . . . 9 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
32		eqid 2763	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
33		ineq1 4113	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34		inxp 5814	. . . . . . . . . . . 12 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
35	33, 34	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
36	35	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
37	32, 36	rexrnmpo 7581	. . . . . . . . 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 292	. . . . . . 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 285	. . . . . 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	eqabdv 2897	. . . . 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 2763	. . . . . 6 ⊢ (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
43	42	rnmpo 7573	. . . . 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 2816	. . . 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 6909	. . 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 2804	. 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		ovex 7473	. . 3 ⊢ (𝑅 ↾t 𝐴) ∈ V
48		ovex 7473	. . 3 ⊢ (𝑆 ↾t 𝐵) ∈ V
49		eqid 2763	. . . 4 ⊢ ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
50	49	txval 23910	. . 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))))
51	47, 48, 50	mp2an 705	. 2 ⊢ ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))
52	46, 51	eqtr4di 2816	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ∩ cin 4092   × cxp 5653  ran crn 5656  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442   ↾_t crest 17617  topGenctg 17634   ×_t ctx 23906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-rest 17619  df-topgen 17640  df-tx 23908

This theorem is used by:  txlly  23982  txnlly  23983  txkgen  23998  cnmpt2res  24023  xkoinjcn  24033  cnmpopc  25276  cnheiborlem  25302  lhop1lem  26360  cxpcn3  27103  raddcn  34628  cvmlift2lem6  36239  cvmlift2lem9  36242  cvmlift2lem12  36245