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Theorem ixpconstg 8155
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 8103 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑚 𝐴) = {𝑓𝑓:𝐴𝐵})
2 vex 3386 . . . . 5 𝑓 ∈ V
32elixpconst 8154 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
43abbi2i 2913 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
51, 4syl6reqr 2850 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
65ancoms 451 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {cab 2783  wf 6095  (class class class)co 6876  𝑚 cmap 8093  Xcixp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-map 8095  df-ixp 8147
This theorem is referenced by:  ixpconst  8156  mapsnf1o  8187  prdshom  16439  pwsbas  16459  frlmip  20439  pttoponconst  21726  xkoptsub  21783  xkopt  21784  tmdgsum2  22225  rrxip  23513  ovnlecvr2  41558
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