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Theorem xpmapen 7986
 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 fveq2 6084 . . . 4 (𝑤 = 𝑧 → (𝑥𝑤) = (𝑥𝑧))
54fveq2d 6088 . . 3 (𝑤 = 𝑧 → (1st ‘(𝑥𝑤)) = (1st ‘(𝑥𝑧)))
65cbvmptv 4668 . 2 (𝑤𝐶 ↦ (1st ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
74fveq2d 6088 . . 3 (𝑤 = 𝑧 → (2nd ‘(𝑥𝑤)) = (2nd ‘(𝑥𝑧)))
87cbvmptv 4668 . 2 (𝑤𝐶 ↦ (2nd ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
9 fveq2 6084 . . . 4 (𝑤 = 𝑧 → ((1st𝑦)‘𝑤) = ((1st𝑦)‘𝑧))
10 fveq2 6084 . . . 4 (𝑤 = 𝑧 → ((2nd𝑦)‘𝑤) = ((2nd𝑦)‘𝑧))
119, 10opeq12d 4338 . . 3 (𝑤 = 𝑧 → ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩ = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
1211cbvmptv 4668 . 2 (𝑤𝐶 ↦ ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩) = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
131, 2, 3, 6, 8, 12xpmapenlem 7985 1 ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1975  Vcvv 3168  ⟨cop 4126   class class class wbr 4573   ↦ cmpt 4633   × cxp 5022  ‘cfv 5786  (class class class)co 6523  1st c1st 7030  2nd c2nd 7031   ↑𝑚 cmap 7717   ≈ cen 7811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-map 7719  df-en 7815 This theorem is referenced by:  rexpen  14738
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