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Theorem elixp2 8835
Description: Membership in an infinite Cartesian product. See df-ixp 8832 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 6590 . . . . 5 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 fveq1 6838 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
32eleq1d 2822 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
43ralbidv 3172 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4anbi12d 631 . . . 4 (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
6 dfixp 8833 . . . 4 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
75, 6elab2g 3630 . . 3 (𝐹 ∈ V → (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
87pm5.32i 575 . 2 ((𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
9 elex 3461 . . 3 (𝐹X𝑥𝐴 𝐵𝐹 ∈ V)
109pm4.71ri 561 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵))
11 3anass 1095 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
128, 10, 113bitr4i 302 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  Vcvv 3443   Fn wfn 6488  cfv 6493  Xcixp 8831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501  df-ixp 8832
This theorem is referenced by:  fvixp  8836  ixpfn  8837  elixp  8838  ixpf  8854  resixp  8867  undifixp  8868  mptelixpg  8869  prdsbasprj  17346  xpsfrnel  17436  xpscf  17439  isssc  17695  isfuncd  17743  funcres2b  17775  dprdw  19780  ptrecube  36045  kelac1  41328  elixpconstg  43241  fvixp2  43355  rrxsnicc  44473  ioorrnopnxrlem  44479  hoiqssbllem1  44795  iinhoiicclem  44846  iunhoiioolem  44848  funcf2lem  46970
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