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Theorem boxriin 8496
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 767 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2 ssel 3964 . . . . . . . 8 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ 𝐵))
32ral2imi 3160 . . . . . . 7 (∀𝑥𝐼 𝐴𝐵 → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
43adantr 481 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵𝑧 Fn 𝐼) → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
54impr 455 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵)
6 eleq2 2905 . . . . . . . . . . . 12 (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐴 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7 eleq2 2905 . . . . . . . . . . . 12 (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐵 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8 simplr 765 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐴)
9 ssel2 3965 . . . . . . . . . . . . 13 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ 𝐵)
109adantr 481 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐵)
116, 7, 8, 10ifbothda 4506 . . . . . . . . . . 11 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
1211ex 413 . . . . . . . . . 10 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1312ral2imi 3160 . . . . . . . . 9 (∀𝑥𝐼 𝐴𝐵 → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1413adantr 481 . . . . . . . 8 ((∀𝑥𝐼 𝐴𝐵𝑧 Fn 𝐼) → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1514impr 455 . . . . . . 7 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
161, 15jca 512 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1716ralrimivw 3187 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
181, 5, 17jca31 515 . . . 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 775 . . . . 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 485 . . . . . . . 8 ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
2120ralimi 3164 . . . . . . 7 (∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
22 ralcom 3358 . . . . . . . 8 (∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
23 iftrue 4475 . . . . . . . . . . . 12 (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2423equcoms 2020 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2524eleq2d 2902 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧𝑥) ∈ 𝐴))
2625rspcva 3624 . . . . . . . . 9 ((𝑥𝐼 ∧ ∀𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧𝑥) ∈ 𝐴)
2726ralimiaa 3163 . . . . . . . 8 (∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
2822, 27sylbi 218 . . . . . . 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 725 . . . . 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 512 . . . 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 797 . . 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 3502 . . . 4 𝑧 ∈ V
3433elixp 8460 . . 3 (𝑧X𝑥𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴))
35 elin 4172 . . . 4 (𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ (𝑧X𝑥𝐼 𝐵𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3633elixp 8460 . . . . 5 (𝑧X𝑥𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
37 eliin 4921 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3837elv 3504 . . . . . 6 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
3933elixp 8460 . . . . . . 7 (𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4039ralbii 3169 . . . . . 6 (∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4138, 40bitri 276 . . . . 5 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4236, 41anbi12i 626 . . . 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 276 . . 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 315 . 2 (∀𝑥𝐼 𝐴𝐵 → (𝑧X𝑥𝐼 𝐴𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))))
4544eqrdv 2823 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 207  wa 396   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  cin 3938  wss 3939  ifcif 4469   ciin 4917   Fn wfn 6346  cfv 6351  Xcixp 8453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iin 4919  df-br 5063  df-opab 5125  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359  df-ixp 8454
This theorem is referenced by:  ptcld  22139  kelac1  39530
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