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Theorem ixpconstg 8501
 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 3413 . . . . 5 𝑓 ∈ V
21elixpconst 8500 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
32abbi2i 2891 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
4 mapvalg 8432 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵m 𝐴) = {𝑓𝑓:𝐴𝐵})
53, 4eqtr4id 2812 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
65ancoms 462 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵m 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ⟶wf 6336  (class class class)co 7156   ↑m cmap 8422  Xcixp 8492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8424  df-ixp 8493 This theorem is referenced by:  ixpconst  8502  mapsnf1o  8534  prdshom  16812  pwsbas  16832  frlmip  20557  pttoponconst  22311  xkoptsub  22368  xkopt  22369  tmdgsum2  22810  rrxip  24104  ovnlecvr2  43660  naryfvalixp  45467
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