Theorem xpmapen 7103

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 5670	. . . 4 ⊢ (𝑤 = 𝑧 → (𝑥‘𝑤) = (𝑥‘𝑧))
5	4	fveq2d 5674	. . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧)))
6	5	cbvmptv 4206	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
7	4	fveq2d 5674	. . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧)))
8	7	cbvmptv 4206	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
9		fveq2 5670	. . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧))
10		fveq2 5670	. . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧))
11	9, 10	opeq12d 3891	. . 3 ⊢ (𝑤 = 𝑧 → ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩ = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
12	11	cbvmptv 4206	. 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 7102	1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

Syntax hints:   ∈ wcel 2203  Vcvv 2813  ⟨cop 3692   class class class wbr 4109   ↦ cmpt 4171   × cxp 4747  ‘cfv 5352  (class class class)co 6050  1^st c1st 6332  2^nd c2nd 6333   ↑_𝑚 cmap 6882   ≈ cen 6973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-en 6976

This theorem is referenced by: (None)