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Theorem xpmapen 9092
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 6848 . . 3 (𝑤 = 𝑧 → (1st ‘(𝑥𝑤)) = (1st ‘(𝑥𝑧)))
54cbvmptv 5219 . 2 (𝑤𝐶 ↦ (1st ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
6 2fveq3 6848 . . 3 (𝑤 = 𝑧 → (2nd ‘(𝑥𝑤)) = (2nd ‘(𝑥𝑧)))
76cbvmptv 5219 . 2 (𝑤𝐶 ↦ (2nd ‘(𝑥𝑤))) = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
8 fveq2 6843 . . . 4 (𝑤 = 𝑧 → ((1st𝑦)‘𝑤) = ((1st𝑦)‘𝑧))
9 fveq2 6843 . . . 4 (𝑤 = 𝑧 → ((2nd𝑦)‘𝑤) = ((2nd𝑦)‘𝑧))
108, 9opeq12d 4839 . . 3 (𝑤 = 𝑧 → ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩ = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
1110cbvmptv 5219 . 2 (𝑤𝐶 ↦ ⟨((1st𝑦)‘𝑤), ((2nd𝑦)‘𝑤)⟩) = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
121, 2, 3, 5, 7, 11xpmapenlem 9091 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3444  cop 4593   class class class wbr 5106  cmpt 5189   × cxp 5632  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  m cmap 8768  cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-en 8887
This theorem is referenced by:  rexpen  16115
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