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Mirrors > Home > ILE Home > Th. List > dfxp3 | GIF version |
Description: Define the cross product of three classes. Compare df-xp 4515. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
Ref | Expression |
---|---|
dfxp3 | ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 171 | . . 3 ⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶)) | |
2 | 1 | dfoprab4 6058 | . 2 ⊢ {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
3 | df-xp 4515 | . 2 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} | |
4 | df-3an 949 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)) | |
5 | 4 | oprabbii 5794 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
6 | 2, 3, 5 | 3eqtr4i 2148 | 1 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 〈cop 3500 {copab 3958 × cxp 4507 {coprab 5743 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-oprab 5746 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: (None) |
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