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Theorem ixpin 6750
Description: The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ixpin X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpin
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 anandi 590 . . . 4 ((𝑓 Fn 𝐴 ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
2 elin 3333 . . . . . . 7 ((𝑓𝑥) ∈ (𝐵𝐶) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
32ralbii 2496 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
4 r19.26 2616 . . . . . 6 (∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
53, 4bitri 184 . . . . 5 (∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶) ↔ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
65anbi2i 457 . . . 4 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)) ↔ (𝑓 Fn 𝐴 ∧ (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
7 vex 2755 . . . . . 6 𝑓 ∈ V
87elixp 6732 . . . . 5 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
97elixp 6732 . . . . 5 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
108, 9anbi12i 460 . . . 4 ((𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
111, 6, 103bitr4i 212 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)) ↔ (𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶))
127elixp 6732 . . 3 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
13 elin 3333 . . 3 (𝑓 ∈ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) ↔ (𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶))
1411, 12, 133bitr4i 212 . 2 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ 𝑓 ∈ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶))
1514eqriv 2186 1 X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wcel 2160  wral 2468  cin 3143   Fn wfn 5230  cfv 5235  Xcixp 6725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ixp 6726
This theorem is referenced by: (None)
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