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Theorem ixpconst 7865
Description: Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.)
Hypotheses
Ref Expression
ixpconst.1 𝐴 ∈ V
ixpconst.2 𝐵 ∈ V
Assertion
Ref Expression
ixpconst X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ixpconst
StepHypRef Expression
1 ixpconst.1 . 2 𝐴 ∈ V
2 ixpconst.2 . 2 𝐵 ∈ V
3 ixpconstg 7864 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
41, 2, 3mp2an 707 1 X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  (class class class)co 6607  𝑚 cmap 7805  Xcixp 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-ixp 7856
This theorem is referenced by:  pwcfsdom  9352  prdsval  16039  wunfunc  16483  wunnat  16540  poimirlem30  33092  poimirlem32  33094  ovnovollem1  40193  ovnovollem2  40194
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