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Theorem elixp2 6937
Description: Membership in an infinite Cartesian product. See df-ixp 6934 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
Assertion
Ref Expression
elixp2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem elixp2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fneq1 5444 . . . . 5 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 fveq1 5669 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
32eleq1d 2301 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
43ralbidv 2542 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4anbi12d 473 . . . 4 (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
6 dfixp 6935 . . . 4 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
75, 6elab2g 2964 . . 3 (𝐹 ∈ V → (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
87pm5.32i 454 . 2 ((𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
9 elex 2825 . . 3 (𝐹X𝑥𝐴 𝐵𝐹 ∈ V)
109pm4.71ri 392 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵))
11 3anass 1009 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
128, 10, 113bitr4i 212 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 1005   = wceq 1398  wcel 2203  wral 2520  Vcvv 2813   Fn wfn 5347  cfv 5352  Xcixp 6933
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ixp 6934
This theorem is referenced by:  fvixp  6938  ixpfn  6939  elixp  6940  ixpf  6955  resixp  6968  mptelixpg  6969  prdsbasprj  13493  xpsfrnel  13555  xpscf  13558  depindlem2  16500
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