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Theorem xpmapen 8677
 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 6671 . . 3 (𝑤 = 𝑧 → (1st ‘(𝑥𝑤)) = (1st ‘(𝑥𝑧)))
54cbvmptv 5165 . 2 (𝑤𝐶 ↦ (1st ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
6 2fveq3 6671 . . 3 (𝑤 = 𝑧 → (2nd ‘(𝑥𝑤)) = (2nd ‘(𝑥𝑧)))
76cbvmptv 5165 . 2 (𝑤𝐶 ↦ (2nd ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
8 fveq2 6666 . . . 4 (𝑤 = 𝑧 → ((1st𝑦)‘𝑤) = ((1st𝑦)‘𝑧))
9 fveq2 6666 . . . 4 (𝑤 = 𝑧 → ((2nd𝑦)‘𝑤) = ((2nd𝑦)‘𝑧))
108, 9opeq12d 4809 . . 3 (𝑤 = 𝑧 → ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩ = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
1110cbvmptv 5165 . 2 (𝑤𝐶 ↦ ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩) = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
121, 2, 3, 5, 7, 11xpmapenlem 8676 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2107  Vcvv 3499  ⟨cop 4569   class class class wbr 5062   ↦ cmpt 5142   × cxp 5551  ‘cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682   ↑m cmap 8399   ≈ cen 8498 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-en 8502 This theorem is referenced by:  rexpen  15573
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