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Theorem boxriin 8981
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 (∀𝑥𝐼 𝐴𝐵X𝑥𝐼 𝐴 = (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 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.
StepHypRef Expression
1 simprl 770 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2 ssel 3976 . . . . . . . 8 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ 𝐵))
32ral2imi 3084 . . . . . . 7 (∀𝑥𝐼 𝐴𝐵 → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
43adantr 480 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵𝑧 Fn 𝐼) → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
54impr 454 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵)
6 eleq2 2829 . . . . . . . . . . . 12 (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐴 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7 eleq2 2829 . . . . . . . . . . . 12 (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐵 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8 simplr 768 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐴)
9 ssel2 3977 . . . . . . . . . . . . 13 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ 𝐵)
109adantr 480 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐵)
116, 7, 8, 10ifbothda 4563 . . . . . . . . . . 11 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
1211ex 412 . . . . . . . . . 10 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1312ral2imi 3084 . . . . . . . . 9 (∀𝑥𝐼 𝐴𝐵 → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1413adantr 480 . . . . . . . 8 ((∀𝑥𝐼 𝐴𝐵𝑧 Fn 𝐼) → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1514impr 454 . . . . . . 7 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
161, 15jca 511 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1716ralrimivw 3149 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
181, 5, 17jca31 514 . . . 4 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
19 simprll 778 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → 𝑧 Fn 𝐼)
20 simpr 484 . . . . . . . 8 ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
2120ralimi 3082 . . . . . . 7 (∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
22 ralcom 3288 . . . . . . . 8 (∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
23 iftrue 4530 . . . . . . . . . . . 12 (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2423equcoms 2018 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2524eleq2d 2826 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧𝑥) ∈ 𝐴))
2625rspcva 3619 . . . . . . . . 9 ((𝑥𝐼 ∧ ∀𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧𝑥) ∈ 𝐴)
2726ralimiaa 3081 . . . . . . . 8 (∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
2822, 27sylbi 217 . . . . . . 7 (∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
2921, 28syl 17 . . . . . 6 (∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
3029ad2antll 729 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
3119, 30jca 511 . . . 4 ((∀𝑥𝐼 𝐴𝐵 ∧ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))) → (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴))
3218, 31impbida 800 . . 3 (∀𝑥𝐼 𝐴𝐵 → ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))))
33 vex 3483 . . . 4 𝑧 ∈ V
3433elixp 8945 . . 3 (𝑧X𝑥𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴))
35 elin 3966 . . . 4 (𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ (𝑧X𝑥𝐼 𝐵𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3633elixp 8945 . . . . 5 (𝑧X𝑥𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
37 eliin 4995 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3837elv 3484 . . . . . 6 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
3933elixp 8945 . . . . . . 7 (𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4039ralbii 3092 . . . . . 6 (∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4138, 40bitri 275 . . . . 5 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4236, 41anbi12i 628 . . . 4 ((𝑧X𝑥𝐼 𝐵𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
4335, 42bitri 275 . . 3 (𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵) ∧ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))))
4432, 34, 433bitr4g 314 . 2 (∀𝑥𝐼 𝐴𝐵 → (𝑧X𝑥𝐼 𝐴𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))))
4544eqrdv 2734 1 (∀𝑥𝐼 𝐴𝐵X𝑥𝐼 𝐴 = (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cin 3949  wss 3950  ifcif 4524   ciin 4991   Fn wfn 6555  cfv 6560  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iin 4993  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ixp 8939
This theorem is referenced by:  ptcld  23622  kelac1  43080
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