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Theorem inixp 34873
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 652 . . . 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 565 . . . . 5 ((𝑓 Fn 𝐴𝑓 Fn 𝐴) ↔ 𝑓 Fn 𝐴)
3 r19.26 3174 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
4 elin 4172 . . . . . . . 8 ((𝑓𝑥) ∈ (𝐵𝐶) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
54bicomi 225 . . . . . . 7 (((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (𝑓𝑥) ∈ (𝐵𝐶))
65ralbii 3169 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
73, 6bitr3i 278 . . . . 5 ((∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶))
82, 7anbi12i 626 . . . 4 (((𝑓 Fn 𝐴𝑓 Fn 𝐴) ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
91, 8bitri 276 . . 3 (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
10 vex 3502 . . . . 5 𝑓 ∈ V
1110elixp 8460 . . . 4 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
1210elixp 8460 . . . 4 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
1311, 12anbi12i 626 . . 3 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
1410elixp 8460 . . 3 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
159, 13, 143bitr4i 304 . 2 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ 𝑓X𝑥𝐴 (𝐵𝐶))
1615ineqri 4183 1 (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) = X𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wcel 2107  wral 3142  cin 3938   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-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: (None)
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