MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  boxriin Structured version   Visualization version   GIF version

Theorem boxriin 7902
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 793 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → 𝑧 Fn 𝐼)
2 ssel 3581 . . . . . . . 8 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ 𝐵))
32ral2imi 2942 . . . . . . 7 (∀𝑥𝐼 𝐴𝐵 → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
43adantr 481 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵𝑧 Fn 𝐼) → (∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴 → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
54impr 648 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵)
6 eleq2 2687 . . . . . . . . . . . 12 (𝐴 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐴 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
7 eleq2 2687 . . . . . . . . . . . 12 (𝐵 = if(𝑥 = 𝑦, 𝐴, 𝐵) → ((𝑧𝑥) ∈ 𝐵 ↔ (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
8 simplr 791 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐴)
9 ssel2 3582 . . . . . . . . . . . . 13 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ 𝐵)
109adantr 481 . . . . . . . . . . . 12 (((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) ∧ ¬ 𝑥 = 𝑦) → (𝑧𝑥) ∈ 𝐵)
116, 7, 8, 10ifbothda 4100 . . . . . . . . . . 11 ((𝐴𝐵 ∧ (𝑧𝑥) ∈ 𝐴) → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
1211ex 450 . . . . . . . . . 10 (𝐴𝐵 → ((𝑧𝑥) ∈ 𝐴 → (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1312ral2imi 2942 . . . . . . . . 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 648 . . . . . . 7 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
161, 15jca 554 . . . . . 6 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
1716ralrimivw 2962 . . . . 5 ((∀𝑥𝐼 𝐴𝐵 ∧ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)) → ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
181, 5, 17jca31 556 . . . 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 801 . . . . 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 477 . . . . . . . 8 ((𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
2120ralimi 2947 . . . . . . 7 (∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → ∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
22 ralcom 3091 . . . . . . . 8 (∀𝑦𝐼𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵))
23 iftrue 4069 . . . . . . . . . . . 12 (𝑥 = 𝑦 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2423equcoms 1944 . . . . . . . . . . 11 (𝑦 = 𝑥 → if(𝑥 = 𝑦, 𝐴, 𝐵) = 𝐴)
2524eleq2d 2684 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧𝑥) ∈ 𝐴))
2625rspcva 3296 . . . . . . . . 9 ((𝑥𝐼 ∧ ∀𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)) → (𝑧𝑥) ∈ 𝐴)
2726ralimiaa 2946 . . . . . . . 8 (∀𝑥𝐼𝑦𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵) → ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴)
2822, 27sylbi 207 . . . . . . 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 764 . . . . 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 554 . . . 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 876 . . 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 3192 . . . 4 𝑧 ∈ V
3433elixp 7867 . . 3 (𝑧X𝑥𝐼 𝐴 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐴))
35 elin 3779 . . . 4 (𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)) ↔ (𝑧X𝑥𝐼 𝐵𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3633elixp 7867 . . . . 5 (𝑧X𝑥𝐼 𝐵 ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ 𝐵))
37 eliin 4496 . . . . . . 7 (𝑧 ∈ V → (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵)))
3833, 37ax-mp 5 . . . . . 6 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))
3933elixp 7867 . . . . . . 7 (𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4039ralbii 2975 . . . . . 6 (∀𝑦𝐼 𝑧X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4138, 40bitri 264 . . . . 5 (𝑧 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵) ↔ ∀𝑦𝐼 (𝑧 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑧𝑥) ∈ if(𝑥 = 𝑦, 𝐴, 𝐵)))
4236, 41anbi12i 732 . . . 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 264 . . 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 303 . 2 (∀𝑥𝐼 𝐴𝐵 → (𝑧X𝑥𝐼 𝐴𝑧 ∈ (X𝑥𝐼 𝐵 𝑦𝐼 X𝑥𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))))
4544eqrdv 2619 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cin 3558  wss 3559  ifcif 4063   ciin 4491   Fn wfn 5847  cfv 5852  Xcixp 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iin 4493  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-ixp 7861
This theorem is referenced by:  ptcld  21339  kelac1  37148
  Copyright terms: Public domain W3C validator