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Theorem elixpconst 7901
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 7900 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
3 ffnfv 6374 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
42, 3bitr4i 267 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1988  wral 2909  Vcvv 3195   Fn wfn 5871  wf 5872  cfv 5876  Xcixp 7893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ixp 7894
This theorem is referenced by:  ixpconstg  7902  sscpwex  16456  psrbaglefi  19353
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