Theorem txrest 23577

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 2735	. . . . . 6 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
2	1	txval 23510	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
3	2	adantr 480	. . . 4 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))))
4	3	oveq1d 7373	. . 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 23512	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6		xpexg 7695	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 × 𝐵) ∈ V)
7		tgrest 23105	. . . 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 597	. . 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 17349	. . . . . . . 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 597	. . . . . . 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 3443	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
12	11	inex1 5261	. . . . . . . . . 10 ⊢ (𝑟 ∩ 𝐴) ∈ V
13	12	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ∩ 𝐴) ∈ V)
14		elrest 17349	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
15	14	ad2ant2r 748	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
16		xpeq1 5637	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑢 × 𝑣) = ((𝑟 ∩ 𝐴) × 𝑣))
17	16	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
18	17	rexbidv 3159	. . . . . . . . . 10 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
19		vex 3443	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
20	19	inex1 5261	. . . . . . . . . . . 12 ⊢ (𝑠 ∩ 𝐵) ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝐵) ∈ V)
22		elrest 17349	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐵 ∈ 𝑌) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
23	22	ad2ant2l 747	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
24		xpeq2 5644	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → ((𝑟 ∩ 𝐴) × 𝑣) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
25	24	eqeq2d 2746	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → (𝑥 = ((𝑟 ∩ 𝐴) × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
26	25	adantl 481	. . . . . . . . . . 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 5355	. . . . . . . . . 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 510	. . . . . . . . 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 5355	. . . . . . . 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 7698	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
31	30	rgen2w 3055	. . . . . . . . 9 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
32		eqid 2735	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
33		ineq1 4164	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34		inxp 5779	. . . . . . . . . . . 12 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
35	33, 34	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
36	35	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
37	32, 36	rexrnmpo 7498	. . . . . . . . 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 289	. . . . . . 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 282	. . . . . 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 2868	. . . . 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 2735	. . . . . 6 ⊢ (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
43	42	rnmpo 7491	. . . . 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 2788	. . . 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 6837	. . 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 2776	. 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 7391	. . 3 ⊢ (𝑅 ↾t 𝐴) ∈ V
48		ovex 7391	. . 3 ⊢ (𝑆 ↾t 𝐵) ∈ V
49		eqid 2735	. . . 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 23510	. . 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 693	. 2 ⊢ ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))
52	46, 51	eqtr4di 2788	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   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∃wrex 3059  Vcvv 3439   ∩ cin 3899   × cxp 5621  ran crn 5624  ‘cfv 6491  (class class class)co 7358   ∈ cmpo 7360   ↾_t crest 17342  topGenctg 17359   ×_t ctx 23506

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-rest 17344  df-topgen 17365  df-tx 23508

This theorem is referenced by:  txlly  23582  txnlly  23583  txkgen  23598  cnmpt2res  23623  xkoinjcn  23633  cnmpopc  24880  cnheiborlem  24911  lhop1lem  25976  cxpcn3  26716  raddcn  34065  cvmlift2lem6  35481  cvmlift2lem9  35484  cvmlift2lem12  35487