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Theorem xpmapen 9185
Description: Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
xpmapen.1 𝐴 ∈ V
xpmapen.2 𝐵 ∈ V
xpmapen.3 𝐶 ∈ V
Assertion
Ref Expression
xpmapen ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))

Proof of Theorem xpmapen
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmapen.1 . 2 𝐴 ∈ V
2 xpmapen.2 . 2 𝐵 ∈ V
3 xpmapen.3 . 2 𝐶 ∈ V
4 2fveq3 6911 . . 3 (𝑤 = 𝑧 → (1st ‘(𝑥𝑤)) = (1st ‘(𝑥𝑧)))
54cbvmptv 5255 . 2 (𝑤𝐶 ↦ (1st ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
6 2fveq3 6911 . . 3 (𝑤 = 𝑧 → (2nd ‘(𝑥𝑤)) = (2nd ‘(𝑥𝑧)))
76cbvmptv 5255 . 2 (𝑤𝐶 ↦ (2nd ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
8 fveq2 6906 . . . 4 (𝑤 = 𝑧 → ((1st𝑦)‘𝑤) = ((1st𝑦)‘𝑧))
9 fveq2 6906 . . . 4 (𝑤 = 𝑧 → ((2nd𝑦)‘𝑤) = ((2nd𝑦)‘𝑧))
108, 9opeq12d 4881 . . 3 (𝑤 = 𝑧 → ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩ = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
1110cbvmptv 5255 . 2 (𝑤𝐶 ↦ ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩) = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
121, 2, 3, 5, 7, 11xpmapenlem 9184 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-en 8986
This theorem is referenced by:  rexpen  16264
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