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Theorem ixpiin 7976
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 4099 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 vex 3234 . . . . . 6 𝑓 ∈ V
3 eliin 4557 . . . . . 6 (𝑓 ∈ V → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶))
42, 3ax-mp 5 . . . . 5 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶)
52elixp 7957 . . . . . 6 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
65ralbii 3009 . . . . 5 (∀𝑦𝐵 𝑓X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
74, 6bitri 264 . . . 4 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
82elixp 7957 . . . . 5 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶))
9 fvex 6239 . . . . . . . . 9 (𝑓𝑥) ∈ V
10 eliin 4557 . . . . . . . . 9 ((𝑓𝑥) ∈ V → ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶))
119, 10ax-mp 5 . . . . . . . 8 ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
1211ralbii 3009 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
13 ralcom 3127 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1412, 13bitri 264 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1514anbi2i 730 . . . . 5 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
168, 15bitri 264 . . . 4 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
171, 7, 163bitr4g 303 . . 3 (𝐵 ≠ ∅ → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶𝑓X𝑥𝐴 𝑦𝐵 𝐶))
1817eqrdv 2649 . 2 (𝐵 ≠ ∅ → 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
1918eqcomd 2657 1 (𝐵 ≠ ∅ → X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  c0 3948   ciin 4553   Fn wfn 5921  cfv 5926  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iin 4555  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ixp 7951
This theorem is referenced by:  ixpint  7977  ptbasfi  21432  iccvonmbllem  41213
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