Theorem boxriin 8882

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 771	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2		ssel 3928	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → ((𝑧‘𝑥) ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵))
3	2	ral2imi 3076	. . . . . . 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 2826	. . . . . . . . . . . 12 ⊢ (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐴 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7		eleq2 2826	. . . . . . . . . . . 12 ⊢ (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐵 ∧ (𝑧‘𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧‘𝑥) ∈ 𝐴)
9		ssel2 3929	. . . . . . . . . . . . 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 4519	. . . . . . . . . . 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 3076	. . . . . . . . 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 3133	. . . . 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 779	. . . . 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 3074	. . . . . . 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 3265	. . . . . . . 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 4486	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
24	23	equcoms 2022	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
25	24	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧‘𝑥) ∈ 𝐴))
26	25	rspcva 3575	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧‘𝑥) ∈ 𝐴)
27	26	ralimiaa 3073	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴)
28	22, 27	sylbi 217	. . . . . . 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 730	. . . . 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 801	. . 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 3445	. . . 4 ⊢ 𝑧 ∈ V
34	33	elixp 8846	. . 3 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐴))
35		elin 3918	. . . 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 8846	. . . . 5 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ 𝐵))
37		eliin 4952	. . . . . . 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 3446	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦 ∈ 𝐼 𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
39	33	elixp 8846	. . . . . . 7 ⊢ (𝑧 ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑧‘𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
40	39	ralbii 3083	. . . . . 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 275	. . . . 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 629	. . . 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 275	. . 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 314	. 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 2735	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ifcif 4480  ∩ ciin 4948   Fn wfn 6488  ‘cfv 6493  Xcixp 8839

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-11 2163  ax-ext 2709

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iin 4950  df-br 5100  df-opab 5162  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ixp 8840

This theorem is referenced by:  ptcld  23561  kelac1  43372