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Theorem ixpiin 8917
Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
ixpiin (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem ixpiin
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 4500 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 eliin 5002 . . . . . 6 (𝑓 ∈ V → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶))
32elv 3480 . . . . 5 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶)
4 vex 3478 . . . . . . 7 𝑓 ∈ V
54elixp 8897 . . . . . 6 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
65ralbii 3093 . . . . 5 (∀𝑦𝐵 𝑓X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
73, 6bitri 274 . . . 4 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
84elixp 8897 . . . . 5 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶))
9 fvex 6904 . . . . . . . . 9 (𝑓𝑥) ∈ V
10 eliin 5002 . . . . . . . . 9 ((𝑓𝑥) ∈ V → ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶))
119, 10ax-mp 5 . . . . . . . 8 ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
1211ralbii 3093 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
13 ralcom 3286 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1412, 13bitri 274 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1514anbi2i 623 . . . . 5 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
168, 15bitri 274 . . . 4 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
171, 7, 163bitr4g 313 . . 3 (𝐵 ≠ ∅ → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶𝑓X𝑥𝐴 𝑦𝐵 𝐶))
1817eqrdv 2730 . 2 (𝐵 ≠ ∅ → 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
1918eqcomd 2738 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  Vcvv 3474  c0 4322   ciin 4998   Fn wfn 6538  cfv 6543  Xcixp 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iin 5000  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ixp 8891
This theorem is referenced by:  ixpint  8918  ptbasfi  23084  iccvonmbllem  45384
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