Theorem xpmapen 8687

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		2fveq3 6677	. . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧)))
5	4	cbvmptv 5171	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
6		2fveq3 6677	. . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧)))
7	6	cbvmptv 5171	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
8		fveq2 6672	. . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧))
9		fveq2 6672	. . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧))
10	8, 9	opeq12d 4813	. . 3 ⊢ (𝑤 = 𝑧 → ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩ = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
11	10	cbvmptv 5171	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩) = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
12	1, 2, 3, 5, 7, 11	xpmapenlem 8686	1 ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶))

Syntax hints:   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690   ↑_m cmap 8408   ≈ cen 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-map 8410  df-en 8512

This theorem is referenced by:  rexpen  15583