Theorem xpmapen 7150

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) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

Proof of Theorem xpmapen

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpmapen.1	. 2 ⊢ 𝐴 ∈ V
2		xpmapen.2	. 2 ⊢ 𝐵 ∈ V
3		xpmapen.3	. 2 ⊢ 𝐶 ∈ V
4		fveq2 5695	. . . 4 ⊢ (𝑤 = 𝑧 → (𝑥‘𝑤) = (𝑥‘𝑧))
5	4	fveq2d 5699	. . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧)))
6	5	cbvmptv 4227	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
7	4	fveq2d 5699	. . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧)))
8	7	cbvmptv 4227	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
9		fveq2 5695	. . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧))
10		fveq2 5695	. . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧))
11	9, 10	opeq12d 3912	. . 3 ⊢ (𝑤 = 𝑧 → ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩ = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
12	11	cbvmptv 4227	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩) = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
13	1, 2, 3, 6, 8, 12	xpmapenlem 7149	1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

This proof depends on syntax axioms:   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712   class class class wbr 4130   ↦ cmpt 4192   × cxp 4772  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373   ↑_𝑚 cmap 6922   ≈ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-en 7023

This theorem is used by: (None)