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Theorem xpmapen 9176
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 6907 . . 3 (𝑤 = 𝑧 → (1st ‘(𝑥𝑤)) = (1st ‘(𝑥𝑧)))
54cbvmptv 5265 . 2 (𝑤𝐶 ↦ (1st ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
6 2fveq3 6907 . . 3 (𝑤 = 𝑧 → (2nd ‘(𝑥𝑤)) = (2nd ‘(𝑥𝑧)))
76cbvmptv 5265 . 2 (𝑤𝐶 ↦ (2nd ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
8 fveq2 6902 . . . 4 (𝑤 = 𝑧 → ((1st𝑦)‘𝑤) = ((1st𝑦)‘𝑧))
9 fveq2 6902 . . . 4 (𝑤 = 𝑧 → ((2nd𝑦)‘𝑤) = ((2nd𝑦)‘𝑧))
108, 9opeq12d 4886 . . 3 (𝑤 = 𝑧 → ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩ = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
1110cbvmptv 5265 . 2 (𝑤𝐶 ↦ ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩) = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
121, 2, 3, 5, 7, 11xpmapenlem 9175 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3473  cop 4638   class class class wbr 5152  cmpt 5235   × cxp 5680  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  m cmap 8851  cen 8967
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-en 8971
This theorem is referenced by:  rexpen  16212
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