Theorem txrest 23552

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 2729	. . . . . 6 ⊢ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
2	1	txval 23485	. . . . 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 7384	. . 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 23487	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6		xpexg 7706	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 × 𝐵) ∈ V)
7		tgrest 23080	. . . 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 596	. . 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 17367	. . . . . . . 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 596	. . . . . . 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 3448	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
12	11	inex1 5267	. . . . . . . . . 10 ⊢ (𝑟 ∩ 𝐴) ∈ V
13	12	a1i 11	. . . . . . . . 9 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ∩ 𝐴) ∈ V)
14		elrest 17367	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
15	14	ad2ant2r 747	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑢 ∈ (𝑅 ↾t 𝐴) ↔ ∃𝑟 ∈ 𝑅 𝑢 = (𝑟 ∩ 𝐴)))
16		xpeq1 5645	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑢 × 𝑣) = ((𝑟 ∩ 𝐴) × 𝑣))
17	16	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
18	17	rexbidv 3157	. . . . . . . . . 10 ⊢ (𝑢 = (𝑟 ∩ 𝐴) → (∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆 ↾t 𝐵)𝑥 = ((𝑟 ∩ 𝐴) × 𝑣)))
19		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
20	19	inex1 5267	. . . . . . . . . . . 12 ⊢ (𝑠 ∩ 𝐵) ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝐵) ∈ V)
22		elrest 17367	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐵 ∈ 𝑌) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
23	22	ad2ant2l 746	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑣 ∈ (𝑆 ↾t 𝐵) ↔ ∃𝑠 ∈ 𝑆 𝑣 = (𝑠 ∩ 𝐵)))
24		xpeq2 5652	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑠 ∩ 𝐵) → ((𝑟 ∩ 𝐴) × 𝑣) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
25	24	eqeq2d 2740	. . . . . . . . . . . 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 5361	. . . . . . . . . 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 5361	. . . . . . . 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 7709	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
31	30	rgen2w 3049	. . . . . . . . 9 ⊢ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V
32		eqid 2729	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))
33		ineq1 4172	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34		inxp 5785	. . . . . . . . . . . 12 ⊢ ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))
35	33, 34	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵)))
36	35	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟 ∩ 𝐴) × (𝑠 ∩ 𝐵))))
37	32, 36	rexrnmpo 7509	. . . . . . . . 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 2861	. . . . 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 2729	. . . . . 6 ⊢ (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣))
43	42	rnmpo 7502	. . . . 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 2782	. . . 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 6844	. . 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 2770	. 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 7402	. . 3 ⊢ (𝑅 ↾t 𝐴) ∈ V
48		ovex 7402	. . 3 ⊢ (𝑆 ↾t 𝐵) ∈ V
49		eqid 2729	. . . 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 23485	. . 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 692	. 2 ⊢ ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅 ↾t 𝐴), 𝑣 ∈ (𝑆 ↾t 𝐵) ↦ (𝑢 × 𝑣)))
52	46, 51	eqtr4di 2782	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 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   × cxp 5629  ran crn 5632  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371   ↾_t crest 17360  topGenctg 17377   ×_t ctx 23481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-rest 17362  df-topgen 17383  df-tx 23483

This theorem is referenced by:  txlly  23557  txnlly  23558  txkgen  23573  cnmpt2res  23598  xkoinjcn  23608  cnmpopc  24856  cnheiborlem  24887  lhop1lem  25952  cxpcn3  26692  raddcn  33913  cvmlift2lem6  35289  cvmlift2lem9  35292  cvmlift2lem12  35295