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Theorem inixp 36237
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 655 . . . 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 566 . . . . 5 ((𝑓 Fn 𝐴𝑓 Fn 𝐴) ↔ 𝑓 Fn 𝐴)
3 r19.26 3111 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
4 elin 3930 . . . . . . . 8 ((𝑓𝑥) ∈ (𝐵𝐶) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
54bicomi 223 . . . . . . 7 (((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (𝑓𝑥) ∈ (𝐵𝐶))
65ralbii 3093 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
73, 6bitr3i 277 . . . . 5 ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
82, 7anbi12i 628 . . . 4 (((𝑓 Fn 𝐴𝑓 Fn 𝐴) ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
91, 8bitri 275 . . 3 (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
10 vex 3451 . . . . 5 𝑓 ∈ V
1110elixp 8848 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1210elixp 8848 . . . 4 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
1311, 12anbi12i 628 . . 3 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
1410elixp 8848 . . 3 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
159, 13, 143bitr4i 303 . 2 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ 𝑓X𝑥𝐴 (𝐵𝐶))
1615ineqri 4168 1 (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) = X𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  wral 3061  cin 3913   Fn wfn 6495  cfv 6500  Xcixp 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-ixp 8842
This theorem is referenced by: (None)
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