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Theorem ixpconstg 8458
Description: Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.)
Assertion
Ref Expression
ixpconstg ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ixpconstg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mapvalg 8405 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵m 𝐴) = {𝑓𝑓:𝐴𝐵})
2 vex 3495 . . . . 5 𝑓 ∈ V
32elixpconst 8457 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
43abbi2i 2950 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
51, 4syl6reqr 2872 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
65ancoms 459 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wf 6344  (class class class)co 7145  m cmap 8395  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ixp 8450
This theorem is referenced by:  ixpconst  8459  mapsnf1o  8491  prdshom  16728  pwsbas  16748  frlmip  20850  pttoponconst  22133  xkoptsub  22190  xkopt  22191  tmdgsum2  22632  rrxip  23920  ovnlecvr2  42769
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