![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpmapen | Structured version Visualization version 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 | 2fveq3 6650 | . . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧))) | |
5 | 4 | cbvmptv 5133 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) |
6 | 2fveq3 6650 | . . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧))) | |
7 | 6 | cbvmptv 5133 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) |
8 | fveq2 6645 | . . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧)) | |
9 | fveq2 6645 | . . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧)) | |
10 | 8, 9 | opeq12d 4773 | . . 3 ⊢ (𝑤 = 𝑧 → 〈((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)〉 = 〈((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)〉) |
11 | 10 | cbvmptv 5133 | . 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 8668 | 1 ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 ↑m cmap 8389 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-en 8493 |
This theorem is referenced by: rexpen 15573 |
Copyright terms: Public domain | W3C validator |