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Theorem elixp2 8117
Description: Membership in an infinite Cartesian product. See df-ixp 8114 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 6157 . . . . 5 (𝑓 = 𝐹 → (𝑓 Fn 𝐴𝐹 Fn 𝐴))
2 fveq1 6374 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
32eleq1d 2829 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
43ralbidv 3133 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4anbi12d 624 . . . 4 (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
6 dfixp 8115 . . . 4 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
75, 6elab2g 3508 . . 3 (𝐹 ∈ V → (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
87pm5.32i 570 . 2 ((𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
9 elex 3365 . . 3 (𝐹X𝑥𝐴 𝐵𝐹 ∈ V)
109pm4.71ri 556 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹X𝑥𝐴 𝐵))
11 3anass 1116 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
128, 10, 113bitr4i 294 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350   Fn wfn 6063  cfv 6068  Xcixp 8113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-ixp 8114
This theorem is referenced by:  fvixp  8118  ixpfn  8119  elixp  8120  ixpf  8135  resixp  8148  undifixp  8149  mptelixpg  8150  prdsbasprj  16398  xpsfrnel  16489  isssc  16745  isfuncd  16790  funcres2b  16822  dprdw  18676  ptrecube  33833  kelac1  38310  elixpconstg  39917  fvixp2  40035  rrxsnicc  41157  ioorrnopnxrlem  41163  hoiqssbllem1  41476  iinhoiicclem  41527  iunhoiioolem  41529
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