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Theorem elixp 6704
Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
Hypothesis
Ref Expression
elixp.1 𝐹 ∈ V
Assertion
Ref Expression
elixp (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem elixp
StepHypRef Expression
1 elixp2 6701 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 elixp.1 . . 3 𝐹 ∈ V
3 3anass 982 . . 3 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
42, 3mpbiran 940 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4bitri 184 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 978  wcel 2148  wral 2455  Vcvv 2737   Fn wfn 5211  cfv 5216  Xcixp 6697
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 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ixp 6698
This theorem is referenced by:  elixpconst  6705  ixpin  6722  ixpiinm  6723  elixpsn  6734
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