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Theorem ixpconstg 7902
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 7852 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑚 𝐴) = {𝑓𝑓:𝐴𝐵})
2 vex 3198 . . . . 5 𝑓 ∈ V
32elixpconst 7901 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
43abbi2i 2736 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
51, 4syl6reqr 2673 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
65ancoms 469 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {cab 2606  wf 5872  (class class class)co 6635  𝑚 cmap 7842  Xcixp 7893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-ixp 7894
This theorem is referenced by:  ixpconst  7903  mapsnf1o  7934  prdshom  16108  pwsbas  16128  frlmip  20098  pttoponconst  21381  xkoptsub  21438  xkopt  21439  tmdgsum2  21881  rrxip  23159  ovnlecvr2  40587
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