Theorem boxriin 8686

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 767	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2		ssel 3910	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((𝑧‘𝑥) ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
3	2	ral2imi 3081	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
4	3	adantr 480	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ 𝑧 Fn 𝐼) → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
5	4	impr 454	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵)
6		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐴 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7		eleq2 2827	. . . . . . . . . . . 12 ⊢ (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8		simplr 765	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧‘𝑥) ∈ 𝐴)
9		ssel2 3912	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) → (𝑧‘𝑥) ∈ 𝐵)
10	9	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → (𝑧‘𝑥) ∈ 𝐵)
11	6, 7, 8, 10	ifbothda 4494	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) → (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
12	11	ex 412	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ((𝑧‘𝑥) ∈ 𝐴 → (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
13	12	ral2imi 3081	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴 → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
14	13	adantr 480	. . . . . . . 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 454	. . . . . . 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 511	. . . . . 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 3108	. . . . 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 514	. . . 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 775	. . . . 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 484	. . . . . . . 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 3086	. . . . . . 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 3280	. . . . . . . 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 4462	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
24	23	equcoms 2024	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
25	24	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧‘𝑥) ∈ 𝐴))
26	25	rspcva 3550	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧‘𝑥) ∈ 𝐴)
27	26	ralimiaa 3085	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
28	22, 27	sylbi 216	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
29	21, 28	syl 17	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
30	29	ad2antll 725	. . . . 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 511	. . . 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 797	. . 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 3426	. . . 4 ⊢ 𝑧 ∈ V
34	33	elixp 8650	. . 3 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴))
35		elin 3899	. . . 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 8650	. . . . 5 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
37		eliin 4926	. . . . . . 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 3428	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
39	33	elixp 8650	. . . . . . 7 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
40	39	ralbii 3090	. . . . . 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 274	. . . . 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 626	. . . 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 274	. . 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 313	. 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 2736	1 ⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ifcif 4456  ∩ ciin 4922   Fn wfn 6413  ‘cfv 6418  Xcixp 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iin 4924  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-ixp 8644

This theorem is referenced by:  ptcld  22672  kelac1  40804