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Theorem elixpconst 7802
 Description: Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
Hypothesis
Ref Expression
elixp.1 𝐹 ∈ V
Assertion
Ref Expression
elixpconst (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elixpconst
StepHypRef Expression
1 elixp.1 . . 3 𝐹 ∈ V
21elixp 7801 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
3 ffnfv 6295 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
42, 3bitr4i 266 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ixp 7795 This theorem is referenced by:  ixpconstg  7803  sscpwex  16298  psrbaglefi  19193
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