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Theorem ixpin 6717
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 3318 . . . . . . 7 ((𝑓𝑥) ∈ (𝐵𝐶) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
32ralbii 2483 . . . . . 6 (∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ∈ 𝐶))
4 r19.26 2603 . . . . . 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 2740 . . . . . 6 𝑓 ∈ V
87elixp 6699 . . . . 5 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
97elixp 6699 . . . . 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 6699 . . 3 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ (𝐵𝐶)))
13 elin 3318 . . 3 (𝑓 ∈ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶) ↔ (𝑓X𝑥𝐴 𝐵𝑓X𝑥𝐴 𝐶))
1411, 12, 133bitr4i 212 . 2 (𝑓X𝑥𝐴 (𝐵𝐶) ↔ 𝑓 ∈ (X𝑥𝐴 𝐵X𝑥𝐴 𝐶))
1514eqriv 2174 1 X𝑥𝐴 (𝐵𝐶) = (X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  wral 2455  cin 3128   Fn wfn 5207  cfv 5212  Xcixp 6692
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ixp 6693
This theorem is referenced by: (None)
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