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Theorem ixpconstg 6682
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 2733 . . . . 5 𝑓 ∈ V
21elixpconst 6681 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓:𝐴𝐵)
32abbi2i 2285 . . 3 X𝑥𝐴 𝐵 = {𝑓𝑓:𝐴𝐵}
4 mapvalg 6633 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑚 𝐴) = {𝑓𝑓:𝐴𝐵})
53, 4eqtr4id 2222 . 2 ((𝐵𝑊𝐴𝑉) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
65ancoms 266 1 ((𝐴𝑉𝐵𝑊) → X𝑥𝐴 𝐵 = (𝐵𝑚 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  {cab 2156  wf 5192  (class class class)co 5851  𝑚 cmap 6623  Xcixp 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-map 6625  df-ixp 6674
This theorem is referenced by:  ixpconst  6683  mapsnf1o  6712
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