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Theorem inixp 32693
Description: Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
inixp (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) = X𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem inixp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 an4 861 . . . 4 (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ ((𝑓 Fn 𝐴𝑓 Fn 𝐴) ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 anidm 674 . . . . 5 ((𝑓 Fn 𝐴𝑓 Fn 𝐴) ↔ 𝑓 Fn 𝐴)
3 r19.26 3046 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
4 elin 3758 . . . . . . . 8 ((𝑓𝑥) ∈ (𝐵𝐶) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
54bicomi 213 . . . . . . 7 (((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (𝑓𝑥) ∈ (𝐵𝐶))
65ralbii 2963 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
73, 6bitr3i 265 . . . . 5 ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
82, 7anbi12i 729 . . . 4 (((𝑓 Fn 𝐴𝑓 Fn 𝐴) ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
91, 8bitri 263 . . 3 (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
10 vex 3176 . . . . 5 𝑓 ∈ V
1110elixp 7801 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1210elixp 7801 . . . 4 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
1311, 12anbi12i 729 . . 3 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
1410elixp 7801 . . 3 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
159, 13, 143bitr4i 291 . 2 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ 𝑓X𝑥𝐴 (𝐵𝐶))
1615ineqri 3768 1 (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) = X𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wcel 1977  wral 2896  cin 3539   Fn wfn 5799  cfv 5804  Xcixp 7794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ixp 7795
This theorem is referenced by: (None)
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