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Theorem ixpconstg 8945
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 vex 3482 . . . . 5 𝑓 ∈ V
21elixpconst 8944 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
32eqabi 2875 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
4 mapvalg 8875 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵m 𝐴) = {𝑓𝑓:𝐴𝐵})
53, 4eqtr4id 2794 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
65ancoms 458 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wf 6559  (class class class)co 7431  m cmap 8865  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ixp 8937
This theorem is referenced by:  ixpconst  8946  mapsnf1o  8978  prdshom  17514  pwsbas  17534  frlmip  21816  pttoponconst  23621  xkoptsub  23678  xkopt  23679  tmdgsum2  24120  rrxip  25438  ovnlecvr2  46566  naryfvalixp  48479
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