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Theorem elixp2 8066
Description: Membership in an infinite Cartesian product. See df-ixp 8063 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 6119 . . . . 5 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 fveq1 6331 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
32eleq1d 2835 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
43ralbidv 3135 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4anbi12d 616 . . . 4 (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
6 dfixp 8064 . . . 4 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
75, 6elab2g 3504 . . 3 (𝐹 ∈ V → (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
87pm5.32i 564 . 2 ((𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
9 elex 3364 . . 3 (𝐹X𝑥𝐴 𝐵𝐹 ∈ V)
109pm4.71ri 550 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵))
11 3anass 1080 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
128, 10, 113bitr4i 292 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351   Fn wfn 6026  cfv 6031  Xcixp 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ixp 8063
This theorem is referenced by:  fvixp  8067  ixpfn  8068  elixp  8069  ixpf  8084  resixp  8097  undifixp  8098  mptelixpg  8099  prdsbasprj  16340  xpsfrnel  16431  isssc  16687  isfuncd  16732  funcres2b  16764  dprdw  18617  ptrecube  33742  kelac1  38159  elixpconstg  39787  fvixp2  39908  rrxsnicc  41037  ioorrnopnxrlem  41043  hoiqssbllem1  41356  iinhoiicclem  41407  iunhoiioolem  41409
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