Theorem xpmapen 6853

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 5517	. . . 4 ⊢ (𝑤 = 𝑧 → (𝑥‘𝑤) = (𝑥‘𝑧))
5	4	fveq2d 5521	. . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧)))
6	5	cbvmptv 4101	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
7	4	fveq2d 5521	. . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧)))
8	7	cbvmptv 4101	. 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
9		fveq2 5517	. . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧))
10		fveq2 5517	. . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧))
11	9, 10	opeq12d 3788	. . 3 ⊢ (𝑤 = 𝑧 → ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩ = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
12	11	cbvmptv 4101	. 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 6852	1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

Syntax hints:   ∈ wcel 2148  Vcvv 2739  ⟨cop 3597   class class class wbr 4005   ↦ cmpt 4066   × cxp 4626  ‘cfv 5218  (class class class)co 5878  1^st c1st 6142  2^nd c2nd 6143   ↑_𝑚 cmap 6651   ≈ cen 6741

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-map 6653  df-en 6744

This theorem is referenced by: (None)