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Theorem ixpconstg 8462
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 8409 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵m 𝐴) = {𝑓𝑓:𝐴𝐵})
2 vex 3502 . . . . 5 𝑓 ∈ V
32elixpconst 8461 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
43abbi2i 2957 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
51, 4syl6reqr 2879 . 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 1530  wcel 2107  {cab 2803  wf 6347  (class class class)co 7151  m cmap 8399  Xcixp 8453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-ixp 8454
This theorem is referenced by:  ixpconst  8463  mapsnf1o  8495  prdshom  16732  pwsbas  16752  frlmip  20840  pttoponconst  22123  xkoptsub  22180  xkopt  22181  tmdgsum2  22622  rrxip  23910  ovnlecvr2  42760
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