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Theorem elixpconst 8156
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 8155 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
3 ffnfv 6614 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
42, 3bitr4i 270 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wcel 2157  wral 3089  Vcvv 3385   Fn wfn 6096  wf 6097  cfv 6101  Xcixp 8148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ixp 8149
This theorem is referenced by:  ixpconstg  8157  sscpwex  16789  psrbaglefi  19695
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