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Mirrors > Home > ILE Home > Th. List > xpmapen | GIF version |
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.) |
Ref | Expression |
---|---|
xpmapen.1 | ⊢ 𝐴 ∈ V |
xpmapen.2 | ⊢ 𝐵 ∈ V |
xpmapen.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
xpmapen | ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpmapen.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | xpmapen.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | xpmapen.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | fveq2 5421 | . . . 4 ⊢ (𝑤 = 𝑧 → (𝑥‘𝑤) = (𝑥‘𝑧)) | |
5 | 4 | fveq2d 5425 | . . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧))) |
6 | 5 | cbvmptv 4024 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) |
7 | 4 | fveq2d 5425 | . . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧))) |
8 | 7 | cbvmptv 4024 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) |
9 | fveq2 5421 | . . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧)) | |
10 | fveq2 5421 | . . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧)) | |
11 | 9, 10 | opeq12d 3713 | . . 3 ⊢ (𝑤 = 𝑧 → 〈((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)〉 = 〈((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)〉) |
12 | 11 | cbvmptv 4024 | . 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 6743 | 1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2686 〈cop 3530 class class class wbr 3929 ↦ cmpt 3989 × cxp 4537 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 2nd c2nd 6037 ↑𝑚 cmap 6542 ≈ cen 6632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-en 6635 |
This theorem is referenced by: (None) |
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