Theorem boxriin 8968

Description: A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
boxriin	⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐼,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem boxriin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 783	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2		ssel 3902	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((𝑧‘𝑥) ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
3	2	ral2imi 3101	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
4	3	adantr 486	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ 𝑧 Fn 𝐼) → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
5	4	impr 460	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵)
6		eleq2 2849	. . . . . . . . . . . 12 ⊢ (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐴 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7		eleq2 2849	. . . . . . . . . . . 12 ⊢ (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧‘𝑥) ∈ 𝐴)
9		ssel2 3903	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) → (𝑧‘𝑥) ∈ 𝐵)
10	9	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → (𝑧‘𝑥) ∈ 𝐵)
11	6, 7, 8, 10	ifbothda 4520	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) → (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
12	11	ex 418	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ((𝑧‘𝑥) ∈ 𝐴 → (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
13	12	ral2imi 3101	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
14	13	adantr 486	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ 𝑧 Fn 𝐼) → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
15	14	impr 460	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
16	1, 15	jca 521	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
17	16	ralrimivw 3158	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
18	1, 5, 17	jca31 524	. . . 4 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
19		simprll 791	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → 𝑧 Fn 𝐼)
20		simpr 490	. . . . . . . 8 ⊢ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
21	20	ralimi 3099	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑦 ∈ 𝐼 ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
22		ralcom 3290	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
23		iftrue 4487	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
24	23	equcoms 2052	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
25	24	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧‘𝑥) ∈ 𝐴))
26	25	rspcva 3574	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧‘𝑥) ∈ 𝐴)
27	26	ralimiaa 3098	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
28	22, 27	sylbi 220	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
29	21, 28	syl 18	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
30	29	ad2antll 742	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
31	19, 30	jca 521	. . . 4 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴))
32	18, 31	impbida 813	. . 3 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))))
33		vex 3454	. . . 4 ⊢ 𝑧 ∈ V
34	33	elixp 8932	. . 3 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴))
35		elin 4093	. . . 4 ⊢ (𝑧 ∈ (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐵 ∧ 𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
36	33	elixp 8932	. . . . 5 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
37		eliin 4955	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
38	37	elv 3455	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
39	33	elixp 8932	. . . . . . 7 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
40	39	ralbii 3108	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐼 𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
41	38, 40	bitri 278	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
42	36, 41	anbi12i 640	. . . 4 ⊢ ((𝑧 ∈ X𝑥 ∈ 𝐼 𝐵 ∧ 𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
43	35, 42	bitri 278	. . 3 ⊢ (𝑧 ∈ (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
44	32, 34, 43	3bitr4g 317	. 2 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → (𝑧 ∈ X𝑥 ∈ 𝐼 𝐴 ↔ 𝑧 ∈ (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))))
45	44	eqrdv 2758	1 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ifcif 4481  ∩ ciin 4951   Fn wfn 6533  ‘cfv 6538  Xcixp 8925

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iin 4953  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-ixp 8926

This theorem is used by:  ptcld  23932  kelac1  44065