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Theorem ixpiin 8920
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 4499 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 eliin 5001 . . . . . 6 (𝑓 ∈ V → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶))
32elv 3478 . . . . 5 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶)
4 vex 3476 . . . . . . 7 𝑓 ∈ V
54elixp 8900 . . . . . 6 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
65ralbii 3091 . . . . 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 8900 . . . . 5 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶))
9 fvex 6903 . . . . . . . . 9 (𝑓𝑥) ∈ V
10 eliin 5001 . . . . . . . . 9 ((𝑓𝑥) ∈ V → ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶))
119, 10ax-mp 5 . . . . . . . 8 ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
1211ralbii 3091 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
13 ralcom 3284 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1412, 13bitri 274 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1514anbi2i 621 . . . . 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 2728 . 2 (𝐵 ≠ ∅ → 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
1918eqcomd 2736 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wne 2938  wral 3059  Vcvv 3472  c0 4321   ciin 4997   Fn wfn 6537  cfv 6542  Xcixp 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iin 4999  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ixp 8894
This theorem is referenced by:  ixpint  8921  ptbasfi  23305  iccvonmbllem  45692
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