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Theorem ixpiin 8475
 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 4418 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 eliin 4899 . . . . . 6 (𝑓 ∈ V → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶))
32elv 3474 . . . . 5 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶)
4 vex 3472 . . . . . . 7 𝑓 ∈ V
54elixp 8455 . . . . . 6 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
65ralbii 3157 . . . . 5 (∀𝑦𝐵 𝑓X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
73, 6bitri 278 . . . 4 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
84elixp 8455 . . . . 5 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶))
9 fvex 6665 . . . . . . . . 9 (𝑓𝑥) ∈ V
10 eliin 4899 . . . . . . . . 9 ((𝑓𝑥) ∈ V → ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶))
119, 10ax-mp 5 . . . . . . . 8 ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
1211ralbii 3157 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
13 ralcom 3335 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1412, 13bitri 278 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1514anbi2i 625 . . . . 5 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
168, 15bitri 278 . . . 4 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
171, 7, 163bitr4g 317 . . 3 (𝐵 ≠ ∅ → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶𝑓X𝑥𝐴 𝑦𝐵 𝐶))
1817eqrdv 2820 . 2 (𝐵 ≠ ∅ → 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
1918eqcomd 2828 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  Vcvv 3469  ∅c0 4265  ∩ ciin 4895   Fn wfn 6329  ‘cfv 6334  Xcixp 8448 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iin 4897  df-br 5043  df-opab 5105  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ixp 8449 This theorem is referenced by:  ixpint  8476  ptbasfi  22184  iccvonmbllem  43256
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